# On the monophonic convexity in complementary prisms

Neethu P. K.1 Ullas Chandran S. V.1 Julliano R. Nascimento2

###### Abstract

A set $S$ of vertices of a graph $G$ is _monophonic convex_ if $S$ contains
all the vertices belonging to any induced path connecting two vertices of $S$.
The cardinality of a maximum proper monophonic convex set of $G$ is called the
_monophonic convexity number_ of $G$. The _monophonic interval_ of a set $S$
of vertices of $G$ is the set $S$ together with every vertex belonging to any
induced path connecting two vertices of $S$. The cardinality of a minimum set
$S\subseteq V(G)$ whose monophonic interval is $V(G)$ is called the
_monophonic number_ of $G$. The _monophonic convex hull_ of a set $S$ of
vertices of $G$ is the smallest monophonic convex set containing $S$ in $G$.
The cardinality of a minimum set $S\subseteq V(G)$ whose monophonic convex
hull is $V(G)$ is called the _monophonic hull number_ of $G$. The
_complementary prism_ $G\overline{G}$ of $G$ is obtained from the disjoint
union of $G$ and its complement $\overline{G}$ by adding the edges of a
perfect matching between them. In this work, we determine the monophonic
convexity number, the monophonic number, and the monophonic hull number of the
complementary prisms of all graphs.

1Department of Mathematics, Mahatma Gandhi College, University of Kerala,
Kesavadasapuram, Thiruvananthapuram-695004, Kerala, India

<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Instituto de Informática, Universidade Federal de Goiás, Goiânia, GO, Brazil
<EMAIL_ADDRESS>
Key words: Complementary prism; monophonic path; convex set; convexity number;
monophonic number; hull number.

AMS Subj. Class: 05C12; 05C69.

##  1 Introduction

A _finite convexity space_ is a pair $(V,{\cal C})$ consisting of a non-empty
finite set $V$ and a collection ${\cal C}$ of subsets of $V$ such that
$\emptyset\in{\cal C}$, $V\in{\cal C}$ and ${\cal C}$ is closed under
intersections. The elements of $\mathcal{C}$ are called _convex sets_ , see
[19]. Different convexities associated with the vertex set of a graph are
well-known. The most natural convexities in graphs are path convexities
defined by a family of paths $\mathcal{P}$, in a way that a set $S$ of
vertices of $G$ is $\mathcal{P}$-_convex_ if $S$ contains all vertices of
every path of $\mathcal{P}$ between vertices of $S$. An extensive survey of
different types of path convexities can be found in [18]. The well-known
_geodetic convexity_ has $\mathcal{P}$ as the family of all shortest paths [1,
2, 12].

In this paper, we consider the _monophonic convexity_ in graphs. In this
convexity, $\cal P$ is based on the family of induced paths. A _chord_ of a
path $P:u_{0},u_{1},\dots,u_{n}$ is an edge $u_{i}u_{j}$, with $j\geq i+2$. A
$u,v$-path $P$ is _monophonic_ (_$m$ -path_, for short) or _induced_ if it is
a chordless path. A set of vertices $S$ of a graph $G$ is a _monophonic convex
set_ ($m$-_convex set_ for short) if $S$ contains all vertices belonging to
any induced path connecting two vertices of $S$. The cardinality of a maximum
proper convex set of $G$ is called the _monophonic convexity number_
($m$-_convexity number_ for short) of $G$, denoted by
$\operatorname{c_{m}}_{m}(G)$.

The monophonic interval $J_{G}[u,v]$ between $u$ and $v$ is the set of
vertices of all $u,v$-monophonic paths in $G$. For $S\subseteq V(G)$ we fix
$J_{G}[S]=\bigcup_{{}_{u,v\in S}}J_{G}[u,v]$. Thus, a set $S$ is _monophonic
convex_ if $J_{G}[S]=S$. A set $S\subseteq V(G)$ is called a _monophonic set_
($m$-set, for short) if $J_{G}[S]=V(G)$. The _monophonic number_ of $G$,
denoted by $\operatorname{m}(G)$, is the cardinality of a minimum monophonic
set of $G$.

The _$m$ -convex hull_ $[S]_{G}$ of a set $S$ in $G$ is the smallest convex
set containing $S$ in $G$. The convex hull $[S]_{G}$ can be formed from the
sequence $J^{p}_{G}[S]$, where $p$ is a non-negative integer,
$J^{0}_{G}[S]=S$, $J^{1}_{G}[S]=J_{G}[S]$, and
$J^{p}_{G}[S]=J_{G}[J_{G}^{p-1}[S]]$, for $p\geq 2$. A set $S\subseteq V(G)$
is called an _$m$ -hull set_ if $[S]_{G}=V(G)$. The _$m$ -hull number_ of $G$,
denoted by $\operatorname{h_{m}}_{m}(G)$, is the cardinality of a minimum
$m$-hull set of $G$.

The monophonic convexity was introduced in 1982, by Jamison [13]. For
additional topics of interest concerning to the monophonic convexity we refer
to the reader [1, 8, 9, 10, 11]. In particular, Dourado, Protti and
Szwarcfiter [8] proved that the decision problems associated with the
monophonic convexity number and the monophonic number are NP-complete on
arbitrary graphs. On the other hand, they prove that the monophonic hull
number of an arbitrary graph can be computed in polynomial time.

If $G$ is a graph and $\overline{G}$ its complement, then the complementary
prism $G\overline{G}$ of $G$ is the graph formed from the disjoint union of
$G$ and $\overline{G}$ by adding the edges of a perfect matching between the
corresponding vertices of $G$ and $\overline{G}$, see [14, 15]. For example,
$C_{5}\overline{C}_{5}$ is the Petersen graph. The geodesic convexity number
of complementary prisms has been investigated recently in [3]. In [3], the
authors proved that the decision problem associated to the convexity number is
NP-complete even restricted to complementary prisms $G\overline{G}$ when both
$G$ and its complement $\overline{G}$ are connected. In the same work, the
authors found the geodetic convexity number of $G\overline{G}$ when $G$ or
$\overline{G}$ is disconnected. In [16], the authors found a formula for the
geodesic convexity number of complementary prisms of all trees, a rich
subclass of the above-stated class. Other related topics studied recently on
complementary prisms include geodetic hull number [4, 7], general position
number [17], and domination [15]. To our knowledge, there is no literature on
the monophonic convexity number of the complementary prisms. In this paper, we
derive formulas for the monophonic convexity number, monophonic number, and
monophonic hull number of the complementary prisms of all graphs. In Section 2
we fix the notations, terminologies and present some initial remarks. In
Sections 3, 4, and 5 we discuss our results on the monophonic convexity
number, monophonic number, and monophonic hull number, respectively.

##  2 Preliminaries

Graphs in this paper are finite, simple, and undirected. For basic graph
terminologies, we follow [5]. Let $G=(V(G),E(G))$ be a graph. For a vertex
$v\in V(G)$, the _open neighborhood_ $N_{G}(v)$ is the set of neighbors of
$v$, while the _closed neighborhood_ $N_{G}[v]=N_{G}(v)\cup\\{v\\}$. The
_degree_ of $v$ is $deg_{G}(v)=|N_{G}(v)|$. A vertex $v$ is _simplicial_ if
$N_{G}(v)$ induces a clique. For $S\subseteq V(G)$, we fix
$N_{G}[S]=\bigcup_{v\in S}N_{G}[v]$; and $N_{G}(S)=(N_{G}[S])\backslash S$. A
vertex $v\in V(G)$ is _pendant_ if $deg_{G}(v)=1$. The _distance_ between two
vertices $u$ and $v$, denoted $d_{G}(u,v)$, is the length of a shortest path
joining them if any; otherwise $d_{G}(u,v)=\infty$. The _eccentricity_ of a
vertex $v\in V(G)$, denoted $ecc_{G}(v)$, is the maximum distance from $v$ to
any vertex in $G$. We may simplify the notations above by omitting the index
$G$ whenever $G$ is clear from the context. The _diameter_ of a graph $G$,
denoted $\operatorname{diam}(G)$, is the maximum eccentricity among the
vertices of $G$. The _order_ of a graph $G$ is the number of vertices in $G$,
denoted by $n(G)$. A connected component $G^{\prime}$ in a disconnected graph
$G$ is called _trivial_ if $n(G^{\prime})=1$, and _non-trivial_ , otherwise.
The _independent number_ $\alpha(G)$ is the cardinality of a largest set of
nonadjacent vertices of $G$. Whereas the _clique number_ $\omega(G)$ is the
cardinality of a largest clique of $G$. For a set $S\subseteq V(G)$, $S^{c}$
denotes the compliment of the set $S$ in $G$. i.e., $S^{c}=V(G)\setminus S$.

If $u\in V(G)\cap V(G\overline{G})$, then the unique neighbour of $u$ in
$V(G\overline{G})\cap V(\overline{G})$ is denoted by $\overline{u}$, called
the partner of $u$ in $\overline{G}$. If $S\subseteq V(G)$, then the set of
all partners of the vertices from $S$ is denoted by $\overline{S}$. For a
positive integer $k$, we use the notation $[k]=\\{1,2,\dots,k\\}$.

Some remarks that arise from monophonic convexity definitions can be found in
[8]. The next three are useful for our purposes.

######  Remark 2.1 ([8])

Every graph with a simplicial vertex satisfies
$\operatorname{c_{m}}_{m}(G)=n-1$.

######  Remark 2.2 ([8])

Every monophonic set or $m$-hull set of a graph $G$ must contain all of its
simplicial vertices.

######  Remark 2.3 ([8])

For any graph $G$,
$\operatorname{h_{m}}_{m}(G)\leq\min\\{\operatorname{m}(G),\operatorname{c_{m}}_{m}(G)+1\\}$.

Furthermore, when dealing with monophonic intervals on complementary prisms,
the next proposition will be useful.

###### Proposition 2.4

Let $G$ be a graph. If $P:u_{1},u_{2},u_{3},u_{4}$ is an induced path in $G$,
then $V(P)\cup V(\overline{P})\subseteq J_{G\overline{G}}[u_{1},u_{4}]$.

Proof.  Given that $u_{1},u_{2},\overline{u_{2}},\overline{u_{4}},u_{4}$ and
$u_{1},\overline{u_{1}},\overline{u_{3}},u_{3},u_{4}$ are monophonic paths in
$G\overline{G}$, the conclusion is direct.

##  3 Monophonic convexity number

In this section, we determine the monophonic convexity number of complementary
prisms $G\overline{G}$. First, we consider the case when both $G$ and
$\overline{G}$ are connected, in Theorem 3.1.

###### Theorem 3.1

Let $G$ be a connected graph such that $\overline{G}$ is connected, then
$\operatorname{c_{m}}_{m}(G\overline{G})=\max\\{\omega(G),\alpha(G)\\}$.

Proof. Let $S$ be a monophonic convex set of $G\overline{G}$ with
$|S|=\operatorname{c_{m}}_{m}(G\overline{G})$ and fix $S_{G}=S\cap V(G)$ and
$S_{\overline{G}}=S\cap V(\overline{G})$. Since $n(G)\geq 2$, we have that
$|S|\geq 2$. Consider the following three cases.
Case 1 : $S_{G}\neq\emptyset$ and $S_{\overline{G}}\neq\emptyset$. In this
case, we prove that $|S|=2$.

First, we prove the following claim.
Claim 1: $S_{\overline{G}}=\overline{S_{G}}$. Let $u\in S_{G}$. Choose $v\in
S_{G}$ such that $\overline{v}\in S_{\overline{G}}$. Let $x\in S_{G}$. Let
$\overline{P}$ be a $\overline{v},\overline{x}$-monophonic path in
$\overline{G}$. Then $\overline{P}$ together with the edge $\overline{x}x$ is
a $x,\overline{v}$-monophonic path containing $\overline{x}$ in
$G\overline{G}$. So $\overline{x}\in S$. Hence $\overline{S_{G}}\subseteq
S_{\overline{G}}$. On the other hand, let $\overline{y}\in S_{\overline{G}}$
and $P$ be a $u,y$-monophonic path in $G$. Then $P$ together with the edge
$y\overline{y}$ is a $u,\overline{y}$–monophonic path containing $y$ in
$G\overline{G}$. Hence $y\in S_{G}$. This shows that
$S_{\overline{G}}\subseteq\overline{S_{G}}$.
Claim 2: $S_{G}$ is a clique or an independent set in $G$. If $|S_{G}|\leq 2$,
then there is nothing to prove. So assume that $|S_{G}|\geq 3$. Suppose there
exists $u,v\in S_{G}$ with $u$ and $v$ adjacent in $G$. Let $w$ be any vertex
distinct from $u$ and $v$ in $G$. Suppose $w$ is nonadjacent to both $u$ and
$v$. Then $\overline{u},\overline{w},\overline{v}$ is an monophonic path in
$G\overline{G}$ containing $\overline{w}$. And hence by Claim 1,
$w,\overline{w}\in S$. Also if $w$ is adjacent to exactly one of $u$ or $v$,
say $u$, then $u,w,\overline{w},\overline{v}$ is an
$u,\overline{v}$-monophonic path in $G\overline{G}$ containing both $w$ and
$\overline{w}$. Hence $w,\overline{w}\in S$. The above observations show that

$[N_{G}(u)\cap N_{G}(v)]^{c}\subseteq S_{G}.$ (1)

Moreover $N_{G}(u)\cap N_{G}(v)\neq\emptyset$, since $S\neq V(G\overline{G})$.
Let $w\in N_{G}(u)\cap N_{G}(v)$. We first show that any vertex $z\in S_{G}$
is adjacent to both $u$ and $v$. This in turn implies that $S_{G}$ is a
clique, since $u$ and $v$ are arbitrary adjacent vertices in $S_{G}$. Suppose
$z$ is non adjacent to $u$. First consider the case $zw\notin E(G)$. Then
$v,w,\overline{w},\overline{z}$ is an $m$-path connecting $v$ and
$\overline{z}$. This shows that $V(G\overline{G})=S$. This is not possible as
$S$ is a proper subset of $G$. Thus $zw\in E(G)$. In this case
$z,w,u,\overline{u}$ is an $m$-path connecting $z$ and $\overline{u}$
containing $w$, and so $w\in S_{G}$. This again shows that
$S=V(G\overline{G})$. Which is not possible. Thus $uz\in E(G)$ and $vz\in
E(G)$. Hence $S_{G}$ must be a clique in $G$ and $S_{\overline{G}}$ is an
independent set in $\overline{G}$.
Now, if $S_{G}$ contains two nonadjacent vertices $u$ and $v$, then replacing
$G$ by $\overline{G}$ in the above arguments we get that $S_{\overline{G}}$ is
a clique.
Note that an independent set in a connected graph is $m$-convex only if it
contains exactly one vertex. Thus in this case $|S|=2$.
Case 2: $S_{\overline{G}}=\emptyset$. Let $u,v\in S_{G}$. If $uv\notin E(G)$,
then $u,\overline{u},\overline{v},v$ is a monophonic path in $G\overline{G}$.
Since $S_{\overline{G}}=\emptyset$, $S_{G}$ must be a clique in $G$. Hence
$|S|\leq\omega(G)$.
Case 3: $S_{G}=\emptyset$. In this case, replacing $G$ by $\overline{G}$ in
Case 1, we get that $|S|\leq\omega(\overline{G})=\alpha(G)$. $\square$

Next, we find the monophonic convexity number of disconnected graphs.

###### Theorem 3.2

Let $G$ be a disconnected graph with components $\\{G_{i}\\}_{i\in[r]}$, and
$k$ be the order of a minimum component of $G$. Then

$\operatorname{c_{m}}_{m}(G\overline{G})=\begin{cases}2n(G)-1\hskip
56.9055pt;$ if $k=1\,,\\\
\max\\{\omega(G),\alpha(G)+2-(\min_{i\in[r]}\\{\alpha(G_{i})\\})\hskip
8.5359pt;$ otherwise$\,.\end{cases}$

Proof. If $k=1$, then the single vertex $v$ in a minimum component of $G$ will
be a simplicial vertex of $G\overline{G}$. Hence
$V(G\overline{G})\setminus\\{v\\}$ is a convex set of $G\overline{G}$ and so
$\operatorname{c_{m}}_{m}(G\overline{G})=2n(G)-1$. Now suppose that $k\geq 2$.
Let $S$ be a monophonic convex set of $G\overline{G}$ with
$|S|=\operatorname{c_{m}}_{m}(G\overline{G})$. Fix $S_{G}=S\cap V(G)$ and
$S_{\overline{G}}=S\cap V(\overline{G})$. Since $k\geq 2$, we have $|S|\geq
2$. Consider the following cases.
Case 1: $S_{\overline{G}}=\emptyset$. Let $u,v\in S_{G}$. If $uv\notin E(G)$,
then $u,\overline{u},\overline{v},v$ is a monophonic path in $G\overline{G}$.
This shows that $\overline{u},\overline{v}\in S$, which is impossible. Thus
$S_{G}$ must be a clique in $G$. Hence $|S|\leq\omega(G)$.
Case 2: $S_{G}=\emptyset$. In this case, replacing $G$ by $\overline{G}$ in
Case 1, we get $|S|\leq\alpha(G)$.
Case 3: $S_{G}\neq\emptyset$ and $S_{\overline{G}}\neq\emptyset$. Let $u\in
S_{G}$ and $\overline{v}\in S_{\overline{G}}$. We first prove the following
two claims.
Claim 1: If $u$ and $v$ are in the same component of $G$, say in $G_{1}$, then

$\overline{S_{G}\cap G_{1}}=S_{\overline{G}}\cap\overline{G_{1}}.$ (2)

Let $x\in S_{G}\cap G_{1}$. Let $\overline{P}$ be a
$\overline{v},\overline{x}$-monophonic path in $\overline{G}$. Then
$\overline{P}$ together with the edge $\overline{x}x$ is a
$x,\overline{v}$-monophonic path containing $\overline{x}$ in $G\overline{G}$.
Hence $\overline{x}\in S$ and so $\overline{S_{G}\cap G_{1}}\subseteq
S_{\overline{G}}\cap\overline{G_{1}}$. On the other hand, let $\overline{y}\in
S_{\overline{G}}\cap\overline{G_{1}}$ and let $P$ be a $u,y$-monophonic path
in $G_{1}$. Then $P$ together with the edge $y\overline{y}$ is a
$u,\overline{y}$-monophonic path containing $y$ in $G\overline{G}$. Hence
$y\in S_{G}$. This shows that
$S_{\overline{G}}\cap\overline{G_{1}}\subseteq\overline{S_{G}\cap G_{1}}$.
Claim 2: For any component, $G_{i}$ of $G$, with $S_{G}\cap
G_{i}\neq\emptyset$ and $S_{\overline{G}}\cap\overline{G_{i}}\neq\emptyset$,

${\rm either\ }|S_{G}\cap G_{i}|=1{\rm\ or\
}|S_{\overline{G_{i}}}\cap\overline{G_{i}}|=1.$ (3)

Suppose that there exist distinct vertices $x_{1},x_{2}$ in $S_{G}\cap G_{i}$
and distinct vertices $\overline{y_{1}}$ and $\overline{y_{2}}$ in
$S_{\overline{G}}\cap\overline{G_{i}}$. Without loss of generality, we may
assume that $i=1$; and also we may assume that $x_{1}\neq y_{1}$. If $x_{1}$
and $y_{1}$ are adjacent vertices in $G$, then $V(\overline{G})\setminus
V(\overline{G_{1}})\subseteq S_{\overline{G}}$. Since
$\overline{x_{1}},\overline{y_{1}}\in S_{\overline{G}}$ by Claim 1 and so
$V(\overline{G})\setminus V(\overline{G_{1}})\subseteq
J_{\overline{G}}[\overline{x_{1}},\overline{y_{1}}]$. Since the order of any
component is greater than 2, we get $V(\overline{G})\subseteq S$. Now recall
that for any pair of adjacent vertices $z$ and $w$ in $V(G_{i})$, with
$i\in[r]$, $\overline{z},z,w,\overline{w}$ is an $m$-path. Since for any $z\in
V(G_{i})$, $\operatorname{deg}_{G}(z)\geq 1$, for any $i\in[r]$,
$V(G_{i})\subseteq S$ as $V(\overline{G_{i}})\subseteq S$. Hence
$V(G\overline{G})\subseteq S$. This is not possible, as $S$ is a proper subset
of $G\overline{G}$. Hence $S_{G}\cap G_{1}$ must be independent. But then $S$
cannot be convex, since $G_{1}$ is connected. Hence the claim follows.

In the following, we prove that $|S_{G}|=1$. Assume the contrary, that there
exist two distinct vertices $u_{1},u_{2}\in S_{G}$. Suppose first that $u_{1}$
and $u_{2}$ are in the same component of $G$, say in $G_{1}$. Then by Claim 1
and Claim 2, we have that $S_{\overline{G}}\cap
V(\overline{G_{1}})=\emptyset$. Choose $\overline{x}\in S_{\overline{G}}$.
Then $\overline{x}\notin\overline{G_{1}}$ and
$u_{1},\overline{u_{1}},\overline{x}$ is a monophonic path in $G\overline{G}$.
Thus this case is not possible as
$S_{\overline{G}}\cap\overline{G_{1}}=\emptyset$. Thus $u_{1}$ and $u_{2}$
must be from different components, say $u_{1}\in G_{1}$ and $u_{2}\in G_{2}$.
Let $x$ be any vertex in $G_{2}$ distinct from $u_{2}$ and let $P$ be a
monophonic path connecting $x$ and $u_{2}$. Then the path
$u_{1},\overline{u_{1}},\overline{x},x$ together with the path $P$ is a
monophonic path connecting $u_{1}$ and $u_{2}$ containing $x$ and
$\overline{x}$. Hence $x,\overline{x}\in S$. Also by Claim 1,
$\overline{u_{2}}\in S$. This contradicts Claim 2. Thus $|S_{G}|=1$.
Claim 3: $S_{\overline{G}}$ is a clique. If possible suppose that
$\overline{x},\overline{y}\in S_{\overline{G}}$, for some $i\in[r]$ with $x$
and $y$ adjacent in $G$. Then the path $\overline{x},x,y,\overline{y}$ is a
monophonic path in $G\overline{G}$. Thus $x,y\in S_{G}$. This is impossible by
Claims 1 and Claim 2. Thus $S_{\overline{G}}$ is a clique.

Let $S_{G}=\\{u\\}$ and without loss of generality we may assume that $u\in
V(G_{1})$. Also, let $l=\min_{j\in[r]}\\{\alpha(G_{j})\\}$. Then by Claim 1
and Claim 2, $|S_{\overline{G}}\cap\overline{G_{i}}|=1$. Thus
$|S|\leq\alpha(G)+2-(\alpha(G_{i}))$ . This shows that $|S|\leq\alpha(G)+2-l$.

Finally, we show that $G\overline{G}$ contains a $m$-convex set of order
$\alpha(G)+2-l$. For that, let $I$ be an independent set in $G$ with
$|I|=\alpha(G)$. Choose a vertex $u$ in $I$ such that $u\in V(G_{i})$ and
$\alpha(G_{i})=l$. Consider the set $H=\\{u,\overline{u}\\}\cup C$, where
$C=\overline{I}\setminus V(\overline{G_{i}})$. Then $|H|=\alpha(G)+2-l$. We
claim that $H$ is a $m$-convex set of $G\overline{G}$. We remark that
$\overline{I}$ is $m$-convex since it is a clique in $G\overline{G}$. Now let
$\overline{x}$ be a vertex in $\overline{I}$ which is distinct from
$\overline{u}$. Then $x$ and $u$ are in different components of $G$ and
$u,\overline{u},\overline{x}$ is the unique $m$-path connecting $u$ and
$\overline{x}$. This shows that the set $H$ must be $m$-convex in
$G\overline{G}$. This in turn implies that
$\operatorname{c_{m}}_{m}(G\overline{G})=\max\\{\omega(G),\alpha(G)+2-l\\}$.
$\square$

##  4 Monophonic number

In this section, we are concerned with the monophonic number of complementary
prisms. As done in Section 3, the results are separated according to the
connectedness of $G$. We begin when $G$ is a disconnected graph.

###### Theorem 4.1

Let $G$ be a disconnected graph with $r\geq 2$ connected components. Then
$\operatorname{m}(G\overline{G})=r$.

Proof.  Let $G_{1},G_{2},\dots,G_{r}$ be the components of $G$. Consider
$S=\\{v_{1},v_{2},\dots,v_{r}\\}$ such that $v_{i}$ is any vertex of
$V(G_{i})$, for every $i\in[r]$. We show that $S$ is a monophonic set. Since
$G_{i}$ is connected, there is a monophonic path $P:v_{i},\dots,u$ in $G_{i}$,
for every $u\in V(G_{i})$. ($P$ can be the shortest path between $v_{i}$ and
$u$, for instance). Then, $P$ together with the edges $u\bar{u}$,
$\bar{u}\bar{v}_{j}$, $\bar{v}_{j}v_{j}$ is a $v_{i},v_{j}$-monophonic path
containing $\\{u,\overline{u}\\}$, for every $u\in V(G_{i})$, for every $1\leq
i<j\leq r$. Then, $J_{G\overline{G}}[S]=V(G\overline{G})$ and
$\operatorname{m}(G\overline{G})\leq r$.

On the other hand, to show that $r$ is the minimum order of a monophonic set
of $G\overline{G}$, we prove Claim 1.

Claim 1: _If $S$ is a monophonic set of $G\overline{G}$, then
$V(G_{i}\overline{G}_{i})\cap S\neq\emptyset$, for every $i\in[r]$._

Let $t\in\\{0,1,\dots,r\\}$ be the number of trivial components in
$G\overline{G}$. If $t\geq 1$ and $1\leq i\leq t$, the conclusion that
$V(G_{i}\overline{G}_{i})\cap S\neq\emptyset$ is direct, since $v_{i}$ is
simplicial. Then, consider that $t+1\leq i\leq r$, for
$t\in\\{0,1,\dots,r-1\\}$.

For a contradiction, suppose that $V(G_{i}\overline{G}_{i})\cap S=\emptyset$,
for some $t+1\leq i\leq r$. Let $v\in V(G_{i})$. Since $S$ is a monophonic
set, $v$ lies in a $x,y$-monophonic path $P$ in $G\overline{G}$, for $x,y\in
V(G\overline{G})\setminus V(G_{i}\overline{G}_{i})$. Since $G_{i}$ is a
nontrivial component and $P$ is an $m$-path, we have that $|V(P)|\geq 6$ as
well as $|V(P)\cap V(\overline{G}_{i})|\geq 2$. Then, let
$P:x,\dots,x^{\prime},x^{\prime\prime},\dots,y^{\prime\prime},y^{\prime},\dots
y$ with $x^{\prime},y^{\prime}\in V(\overline{G})\setminus
V(\overline{G}_{i})$, $x^{\prime\prime},y^{\prime\prime}\in
V(\overline{G}_{i})$, and
$x^{\prime}x^{\prime\prime},y^{\prime}y^{\prime\prime}\in E(\overline{G})$.
Notice that $x^{\prime}y^{\prime\prime}$ is an edge in $\overline{G}$. Since
$x^{\prime}$ and $y^{\prime\prime}$ are not consecutive vertices in $P$, then
$x^{\prime}y^{\prime\prime}$ is a chord, a contradiction.

By Claim 1, we conclude that $\operatorname{m}(G\overline{G})\geq r$ and the
proof is done. $\square$

In the following, we consider both $G$ and $\overline{G}$ be connected graphs.
Recall that $n(G)\geq 4$.

###### Lemma 4.2

Let $G$ be a connected graph such that $\overline{G}$ is connected. If
$\operatorname{diam}(G)\geq 3$, then $\operatorname{m}(G\overline{G})=2$.

Proof.  Let $u,v\in V(G)$ such that $d_{G}(u,v)=3$ and $S=\\{u,v\\}$. Since
$N_{G}(u)\cap N_{G}(v)=\emptyset$, we have that, for every $x\in N_{G}(u)$,
$\overline{x}\in N_{\overline{G}}(\overline{v})$. Then,
$u,x,\overline{x},\overline{v},v$ is a monophonic path and $x,\overline{x}\in
J_{G\overline{G}}[u,v]$. Similarly, for every $y\in N_{G}(v)$, it holds that
$v,y,\overline{y},\overline{u},u$ is a monophonic path and $y,\overline{y}\in
J_{G\overline{G}}[u,v]$.

Next, we show that for every $z\in V(G)\setminus(N_{G}(u)\cup N_{G}(v))$, the
vertices $z,\overline{z}$ lie in an $u,v$-monophonic path. For that, we choose
$w\in\\{u,v\\}$ such that $d_{G}(z,w)$ is minimum and let $P$ be a shortest
$z,w$-path in $G$. Let $w^{\prime}\in\\{u,v\\}\setminus\\{w\\}$. We show that
$P$ together with the edges
$\\{z\overline{z},\bar{z}\bar{w}^{\prime},\overline{w}^{\prime}w^{\prime}\\}$
is an $u,v$-monophonic path in $G\overline{G}$.

Given that $P$ is a shortest path in $G$, as well as in $G\overline{G}$, it
holds that $P$ is a monophonic path in $G\overline{G}$. Notice that $P$ with
$\\{z\overline{z},\bar{z}\bar{w}^{\prime}\\}$ is also a monophonic path in
$G\overline{G}$, since $\bar{z}\notin N_{G\overline{G}}(P\setminus\\{z\\})$
and $\bar{w}^{\prime}\notin N_{G\overline{G}}(P)$. Recall that
$w^{\prime}\notin N_{G\overline{G}}(\overline{z})$, then, remains to show that
$w^{\prime}\notin N_{G\overline{G}}(P)$. That conclusion holds since
$N_{G}(u)\cap N_{G}(v)=\emptyset$ and $P$ is a shortest $z,w$-path in $G$
implies that $V(P)\cap N_{G}(w^{\prime})=\emptyset$.

Hence, $z,\overline{z}\in J_{G\overline{G}}[u,v]$ and $S$ is a monophonic set
of $G\overline{G}$. $\square$

Recall that Lemma 4.2 treats the case $\operatorname{diam}(G)\geq 3$ or
$\operatorname{diam}(\overline{G})\geq 3$. Then, from now on, we consider
$\operatorname{diam}(G)=\operatorname{diam}(\overline{G})=2$. With such a
restriction, the following remark will be useful.

######  Remark 4.3 ([6])

Let $G$ be a graph. If
$\operatorname{diam}(G)=\operatorname{diam}(\overline{G})=2$, then for every
edge $uv\in E(G)$, $ecc_{G}(u)=ecc_{G}(v)=2$ and $V(G)\setminus(N_{G}(u)\cup
N_{G}(v))\neq\emptyset$.

###### Lemma 4.4

Let $G$ be a graph. If $G=C_{5}$, then $\operatorname{m}(G\overline{G})=3$.

Proof.  Let $G$ be a cycle graph on $5$ vertices, with
$V(G)=\\{u_{1},u_{2},\dots,u_{5}\\}$ and $E(G)=\\{u_{i}u_{i+1}:1\leq i\leq
4\\}\cup\\{u_{5}u_{1}\\}$. Consider $S=\\{u_{1},u_{4},\overline{u}_{5}\\}$.
Since $u_{1},u_{5},u_{4}$ is an $m$-path in $G$, $\overline{u}_{5}\in S$, and
by Proposition 2.4, for every $i\in[4]$, $u_{i},\overline{u}_{i}\in
J_{G\overline{G}}[u_{1},u_{4}]$, we conclude that $S$ is a monophonic set of
$G\overline{G}$.

Next, we show that $\operatorname{m}(G\overline{G})\geq 3$. Suppose, by
contradiction, that $\operatorname{m}(G\overline{G})=2$. Let $S=\\{x,y\\}$ be
a monophonic set of $G\overline{G}$. Since $|V(G)|>1$, we have that $xy\notin
E(G\overline{G})$. We consider two cases: _(i)_ $x,y\in V(G)$ and _(ii)_ $x\in
V(G)$ and $y\in V(\overline{G})$.

_(i)_ Since for every pair of nonadjacent vertices $x,y\in V(G)$,
$d_{G}(x,y)=2$, we may assume w.l.o.g. that $x=u_{1}$ and $y=u_{4}$. Since $S$
is an $m$-set of $G\overline{G}$, $\overline{u}_{5}$ lies in a monophonic path
$P:u_{1},\dots,\overline{u}_{5},\dots,u_{4}$. It is clear that $u_{5}\notin
V(P)$, otherwise an edge from
$\\{u_{1},u_{5},u_{4}u_{5},u_{5}\overline{u}_{5}\\}$ would be a chord. This
implies that
$\overline{u}_{2}\overline{u}_{5},\overline{u}_{3}\overline{u}_{5}\in E(P)$.
If $u_{2}\in V(P)$, then $u_{1}u_{2},u_{2}\overline{u}_{2}\in E(P)$. Then,
$u_{3}\notin V(P)$, consequently $u_{4}\overline{u}_{4}\in E(P)$. But
$\overline{u}_{2}\overline{u}_{4}$ is a chord in $P$, a contradiction. So,
consider that $u_{2}\notin V(P)$. Then,
$u_{1}\overline{u}_{1},\overline{u}_{1}\overline{u}_{3}\in E(P)$. This implies
that $\overline{u}_{4}\notin V(P)$, consequently $u_{3}\in V(P)$ as
$u_{4}u_{3},u_{3}\overline{u}_{3}\in E(P)$. But
$\overline{u}_{1}\overline{u}_{3}$ is a chord in $P$, contradiction. Hence,
$\overline{u}_{5}\notin J_{G\overline{G}}[u_{1},u_{4}]$, a contradiction.

_(ii)_ Let $x\in V(G)$ and $y\in V(\overline{G})$ with $xy\notin E(G)$. Since
$d_{G\overline{G}}(x,y)=2$, we may assume w.l.o.g. that $x=u_{1}$ and
$y=\overline{u}_{2}$. Since $S$ is an $m$-set of $G\overline{G}$, $u_{3}$ lies
in a monophonic path $P:u_{1},\dots,u_{3},\dots,u_{4}$. Notice that
$u_{2}\notin V(P)$, otherwise an edge from
$\\{u_{1},u_{2},u_{2}u_{3},u_{2}\overline{u}_{2}\\}$ would be a chord. Then
$u_{3}u_{4},u_{3}\overline{u}_{3}\in E(P)$. If $u_{5}\in V(P)$, then
$u_{1}u_{5},u_{4}u_{5}\in E(P)$. Consequently $\overline{u}_{5}\notin V(P)$
and $\overline{u}_{2}\overline{u}_{4}\in E(P)$. But in this case,
$u_{4}\overline{u}_{4}$ is a chord of $P$, a contradiction. Then, consider
$u_{5}\notin V(P)$. This implies that $u_{1}\overline{u}_{1}\in E(P)$, but
$\overline{u}_{1}\overline{u}_{3}$ is a chord of $P$, also a contradiction.
Therefore, $\operatorname{m}(G\overline{G})\geq 3$. $\square$

###### Lemma 4.5

Let $G$ be a graph such that
$\operatorname{diam}(G)=\operatorname{diam}(\overline{G})=2$. If $G\neq
C_{5}$, then $\operatorname{m}(G\overline{G})=2$.

Proof.  Let $G\neq C_{5}$ be graph with
$\operatorname{diam}(G)=\operatorname{diam}(\overline{G})=2$. We present two
claims for the proof.

Claim 1: _If there exists $uv\in E(G)$ such that $|N_{G}(u)\setminus
N_{G}(v)|\geq 2$, then $\operatorname{m}(G\overline{G})=2$._

Let $uv\in E(G)$ such that $N_{G}(u)\setminus N_{G}(v)=\\{u_{1},u_{2}\\}$ and
$S=\\{u,\overline{v}\\}$. Given that $u,v,\overline{v}$ is an $m$-path, the
conclusion that $v\in J_{G\overline{G}}[S]$ is immediate. By Remark 4.3, we
know that there exists $x\in V(G)\setminus(N_{G}(u)\cup N_{G}(v))$. Then,
$\overline{u},\overline{v}\in N_{\overline{G}}(\overline{x})$. This implies
that $u,\overline{u},\overline{x},\overline{v}$ is an $m$-path and
$\overline{u}\in J_{G\overline{G}}[S]$.

Recall that $u_{1},u_{2}\notin N_{G}(v)$. Then
$\overline{u}_{1},\overline{u}_{2}\in N_{\overline{G}}(\overline{v})$.
Consequently, $u,u_{i},\overline{u}_{i},\overline{v}$, for every
$i\in\\{1,2\\}$, is a monophonic path in $G\overline{G}$ and
$u_{i},\overline{u}_{i}\in J_{G\overline{G}}[S]$. This in turn implies that
$(N_{G}(u)\setminus N_{G}(v))\cup(\overline{N_{G}(u)\setminus
N_{G}(v)})\subseteq J_{G\overline{G}}[S]$.

Now, let $v_{1}\in N_{G}(v)\setminus N_{G}(u)$. Notice that there is no
monophonic $u,v_{1}$-path containing both $u_{1},u_{2}$, otherwise either
$uu_{1}$ or $uu_{2}$ would be a chord. In addition, since
$d_{G}(v_{1},u_{i})\leq 2$, for every $i\in\\{1,2\\}$, a shortest
$v_{1},u_{i}$-path in $G$ does not contain $v$. Then, we select
$i\in\\{1,2\\}$ such that $v_{1},\dots,u_{i}$ is a shortest path in $G$ of
minimum length. Thus,
$u,u_{i},\dots,v_{1},\overline{v}_{1},\overline{u}_{3-i},\overline{v}$ is a
monophonic $u,\overline{v}$-path in $G\overline{G}$. Consequently
$v_{1},\overline{v}_{1}\in J_{G\overline{G}}[S]$. Furthermore, this implies
that $(N_{G}(v)\setminus N_{G}(u))\cup(\overline{N_{G}(v)\setminus
N_{G}(u)})\subseteq J_{G\overline{G}}[S]$.

Next, let $y\in N_{G}(u)\cap N_{G}(v)$. By Remark 4.3, there exists $w\in
V(G)\setminus(N_{G}(u)\cup N_{G}(y))$. If $w\notin N_{G}(v)$, then
$\overline{w}\in N_{\overline{G}}(\overline{v})$, consequently
$u,y,\overline{y},\overline{w},\overline{v}$ is an $m$-path in
$G\overline{G}$. Otherwise, $vw$ being an edge in $G$, Remark 4.3 implies that
there exists $w^{\prime}\in V(G)\setminus(N_{G}(v)\cup N_{G}(w))$. Thus,
$\overline{v},\overline{w}\in N_{\overline{G}}(\overline{w}^{\prime})$. Hence,
$u,y,\overline{y},\overline{w},\overline{w}^{\prime},\overline{v}$ is an
$m$-path in $G\overline{G}$. In both cases, we conclude that
$y,\overline{y}\in J_{G\overline{G}}[S]$, which implies that $(N_{G}(u)\cap
N_{G}(v))\cup(\overline{N_{G}(u)\cap N_{G}(v)})\subseteq
J_{G\overline{G}}[S]$.

Finally, let $x\in V(G)\setminus(N_{G}(u)\cup N_{G}(v))$. Since
$d_{G}(u,x)=2$, there exists $u^{\prime}\in N_{G}(x)$ such that
$u^{\prime}x\in E(G)$. Then, $u,u^{\prime},x,\overline{x},\overline{v}$ is an
$m$-path in $G\overline{G}$. Thus $x,\overline{x}\in J_{G\overline{G}}[S]$ and
$(V(G)\setminus(N_{G}(u)\cup
N_{G}(v)))\cup(\overline{V(G)\setminus(N_{G}(u)\cup N_{G}(v))})\subseteq
J_{G\overline{G}}[S]$.

By the above, $S=\\{u,\overline{v}\\}$ is an $m$-set of $G\overline{G}$,
therefore $\operatorname{m}(G\overline{G})=2$.

Recall that Claim 1 considers the existence of $uv\in E(G)$ such that
$|N_{G}(u)\setminus N_{G}(v)|\geq 2$. Then, for the rest of the proof, we
consider that there is no such an edge, i.e., for every $uv\in E(G)$,
$|N_{G}(u)\setminus N_{G}(v)|\leq 1$. We proceed with Claim 2.

Claim 2: _If there exists a clique of order $3$ in $G$, then
$\operatorname{m}(G\overline{G})=2$._

Let $C=\\{a,b,c\\}$ be a clique in $G$. Since $ab\in E(G)$, Remark 4.3,
implies that there exists $x\in V(G)\setminus(N_{G}(a)\cup N_{G}(b))$. Since
$\operatorname{diam}(G)=2$, there exists $y\in N_{G}(a)\cup N_{G}(b)$ such
that $xy\in E(G)$.

Recall that $ax,bx\notin E(G)$. If $y=c$, then $cx$ is an edge with
$|N_{G}(c)\setminus N_{G}(x)|\geq 2$ and the conclusion follows by Claim 1.
Otherwise, w.l.o.g., suppose that $y\in N_{G}(a)\setminus N_{G}(b)$. If
$cy\notin E(G)$, given that $by\notin E(G)$, we have that $ay$ is an edge with
$|N_{G}(a)\setminus N_{G}(y)|\geq 2$ and, again, the conclusion follows by
Claim 1. So, suppose that $cy\in E(G)$. Since $ax,cx\notin E(G)$, $yx$ is an
edge with $|N_{G}(y)\setminus N_{G}(x)|\geq 2$. Once applying Claim 1, the
conclusion holds.

By Claim 2, we may assume that $G$ contains no $K_{3}$ as an induced subgraph,
that is, $G$ is _triangle-free_. Recall that for every $uv\in E(G)$,
$|N_{G}(u)\setminus N_{G}(v)|\leq 1$. If $G$ is triangle-free and there exists
$uv\in E(G)$, with $|N_{G}(u)\setminus N_{G}(v)|=0$, then $deg_{G}(u)=1$.
Since $\operatorname{diam}(G)=2$, for every $x\in V(G)\setminus\\{u,v\\}$,
$d_{G}(u,x)=2$. But this implies that $N_{G}(u)\cup N_{G}(v)=V(G)$, which
contradicts Remark 4.3. Then $G$ has no pendant vertices. Finally, since for
every $uv\in E(G)$, $|N_{G}(u)\setminus N_{G}(v)|=1$,
$\operatorname{diam}(G)=\operatorname{diam}(\overline{G)}=2$ implies that
$G=C_{5}$, which was proved in Lemma 4.4. $\square$

Putting everything together, we close this section with Theorem 4.6.

###### Theorem 4.6

Let $G$ be a connected graph such that $\overline{G}$ is connected, then

$\operatorname{m}(G\overline{G})=\begin{cases}3\hskip 11.38092pt;$ if
$G=C_{5}\,,\\\ 2\hskip 11.38092pt;$ otherwise$\,.\end{cases}$

Proof.  The conclusion follows by Lemmas 4.2, 4.4, and 4.5. $\square$

##  5 Monophonic hull number

In this section, we determine the monophonic hull number of complementary
prisms. As mentioned in Introduction, in [8] the authors prove that the
monophonic hull number of an arbitrary graph can be computed in polynomial
time. Although they presented an $O(n^{3}m)$-time algorithm for computing such
a parameter on a graph $G$ with $n$ vertices and $m$ edges, we present here
closed formulas for $\operatorname{h_{m}}_{m}(G\overline{G})$ that depend
solely on the number of trivial components of $G$. This leads to a linear time
algorithm for computing the hull number of complementary prisms.

###### Theorem 5.1

Let $G$ be a disconnected graph with $t$ trivial components. Then

$\operatorname{h_{m}}_{m}(G\overline{G})=\begin{cases}2\hskip 28.45274pt;$ if
$t=0\,,\\\ t+1\hskip 11.38092pt;$ otherwise$\,.\end{cases}$

Proof.  Suppose that $G$ has no trivial components. Denote by
$G_{1},G_{2},\dots,G_{r}$ the components of $G$. Let
$S=\\{\overline{u},\overline{v}\\}$ such that $uv\in E(G_{1})$. We show that
$S$ is an $m$-hull set of $G\overline{G}$. Since $\bar{u}\bar{v}\notin
E(\overline{G})$ and, for every $w\notin V(G_{1})$,
$\bar{u}\bar{w},\bar{v}\bar{w}\in E(\overline{G})$, we have that
$\bar{u},\bar{w},\bar{v}$ is a monophonic path in $\overline{G}$. Then
$\bigcup_{i=2}^{r}V(\overline{G}_{i})\subseteq
J_{G\overline{G}}[\overline{u},\overline{v}]$. Since $G_{2}$ is nontrivial,
there exists $x,y\in V(G_{2})$ such that $xy\in E(G)$. By the same argument,
we get $V(\overline{G}_{1})\subseteq
J_{G\overline{G}}[\overline{x},\overline{y}]$. Finally, given that
$V(\overline{G})\subseteq[S]_{G\overline{G}}$ and, for every $i\in[r]$, there
exist $x,y\in V(G_{i})$ such that $xy\in E(G)$, $x,y\in
J_{G\overline{G}}[\overline{x},\overline{y}]$. By applying Claim 1 of proof of
Theorem 3.2 we get that $V(G)\subseteq[S]_{G\overline{G}}$.

Now, suppose that $G$ has $t\geq 1$ trivial components. Let
$G_{1},G_{2},\dots,G_{r}$ denote the components of $G$, where
$V(G_{i})=\\{v_{i}\\}$, for $1\leq i\leq t$, and $n(G_{i})\geq 2$, for
$t+1\leq i\leq r$. Notice that $v_{i}$ is a simplicial vertex, for every
$1\leq i\leq t$. Then, Remark 2.2 implies that $v_{1},v_{2},\dots,v_{t}$
belongs to any $m$-hull set of $G\overline{G}$ and
$\operatorname{h_{m}}_{m}(G\overline{G})\geq t$. For every $i,j\in[t]$, we
know that $\overline{v}_{i}\overline{v}_{j}$ is a chord for every
$v_{i},v_{j}$-path in $G\overline{G}$. Then,
$[\\{v_{1},v_{2},\dots,v_{t}\\}]_{G\overline{G}}=\\{v_{i},\overline{v}_{i}:1\leq
i\leq t\\}$, which implies that $\operatorname{h_{m}}_{m}(G\overline{G})\geq
t+1$.

Let $u$ be any vertex in a nontrivial component of $G$, say $G_{t+1}$. We show
that $S=\\{u,v_{1},v_{2},\dots,v_{t}\\}$ is an $m$-hull set of
$G\overline{G}$. We know that $u,\overline{u},\overline{v}_{i},v_{i}$ (resp.
$u^{\prime},\overline{u}^{\prime},\overline{v}_{i},v_{i}$) is a monophonic
path between $u$ (resp. $u^{\prime}$, for $u^{\prime}\in N_{G}(u)$) and
$v_{i}$, for every $i\in[t]$. Then $(\\{v_{i},\overline{v}_{i}:1\leq i\leq
t\\}\cup N_{G}[u]\cup\overline{N_{G}[u]})\subseteq[S]_{G\overline{G}}$. Let
$u^{\prime}\in N_{G}(u)$. Since $\overline{u}\overline{u^{\prime}}\notin
E(G)$, we have that $\bigcup_{i=t+2}^{r}V(\overline{G}_{i})\subseteq
J_{G\overline{G}}[\overline{u},\overline{u^{\prime}}]$. Given that $G_{t+2}$
is nontrivial, $V(\overline{G}_{t+1})\subseteq
J_{G\overline{G}}[V(\overline{G}_{t+2})]$, which implies that
$V(\overline{G})\subseteq[S]_{G\overline{G}}$. Finally, using a similar
argument as in the case of $t=0$, we conclude that
$V(G)\subseteq[S]_{G\overline{G}}$. $\square$

###### Theorem 5.2

Let $G$ be a connected graph such that $\overline{G}$ is connected, then
$\operatorname{h_{m}}_{m}(G\overline{G})=2$.

Proof.  Since $\operatorname{h_{m}}_{m}(G\overline{G})$ is upper bounded by
$\operatorname{m}(G\overline{G})$ (cf. Remark 2.3), when $G\neq C_{5}$,
Theorem 4.6 imples that $\operatorname{h_{m}}_{m}(G\overline{G})=2$. Then, it
is enough to prove that $\operatorname{h_{m}}_{m}(C_{5}\overline{C}_{5})=2$.

Let $G$ be a cycle graph on $5$ vertices, with
$V(G)=\\{u_{1},u_{2},\dots,u_{5}\\}$ and $E(G)=\\{u_{i}u_{i+1}:1\leq i\leq
4\\}\cup\\{u_{5}u_{1}\\}$. Consider $S=\\{u_{1},u_{4}\\}$. Proposition 2.4
implies that, for every $i\in[4]$, $u_{i},\overline{u}_{i}\in
J_{G\overline{G}}[S]\subseteq[S]_{G\overline{G}}$. Since $u_{1},u_{5},u_{4}$
is an $m$-path in $G$, $u_{5}\in
J_{G\overline{G}}[S]\subseteq[S]_{G\overline{G}}$. Finally,
$\overline{u}_{2},\overline{u}_{5},\overline{u}_{3}$ is an $m$-path in
$G\overline{G}$, then $\overline{u}_{5}\in
J_{G\overline{G}}[\overline{u}_{2},\overline{u}_{3}]\subseteq
J_{G\overline{G}}^{2}[S]\subseteq[S]_{G\overline{G}}$. Therefore $S$ is an
$m$-hull set of $G\overline{G}$ and
$\operatorname{h_{m}}_{m}(G\overline{G})=2$. $\square$

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11institutetext: Khaled Said (✉) 22institutetext: School of Mathematics and
Physics, University of Queensland, Brisbane, QLD 4072, Australia, 22email:
<EMAIL_ADDRESS>
# Tully-Fisher relation

Khaled Said

###### Abstract

The observed radial velocity of a galaxy consists of two main components: the
recession velocity caused by the smooth Hubble expansion and the peculiar
velocity resulting from the gravitational attraction of growing structures due
to matter density fluctuations. To isolate the recession velocity component
and calculate the Hubble constant, accurate measurements of true distances are
needed. The Tully-Fisher relation is an empirical correlation between the
luminosity and rotational velocity of spiral galaxies that serves as a
distance indicator to measure distances independent of redshift. The Tully-
Fisher relation has played an important role in Hubble constant measurements
since its inception. This chapter delves into the significance of the Tully-
Fisher relation in such measurements and explores its implications. We begin
by discussing the definition and historical background of the Tully-Fisher
relation. We also explore the observational evidence supporting this relation
and discuss its advantages and limitations. The chapter then focuses on the
methodology of using the Tully-Fisher relation for Hubble constant
measurements. This includes detailed explanations of calibration techniques
and biases. We emphasize the advantages of utilizing the Tully-Fisher
relation, such as its ability to provide accurate distance measurements even
at significant redshift where other methods may encounter challenges.

##  0.1 Introduction

###  0.1.1 Description of the Tully-Fisher relation

The Tully-Fisher (TF) relation is an empirical relation that correlates the
intrinsic brightness of a spiral galaxy, measured by its total luminosity, and
its dynamical properties, measured by its rotational velocity. In 1977, Brent
Tully and Richard Fisher proposed the use of this relation as a distance
indicator based on their observations of spiral galaxies in the Virgo cluster
Tully-Fisher1977 . The main idea behind distance indicators is to use
distance-independent observables to predict a distance-dependent parameter,
which can subsequently be compared with the corresponding observable to derive
a distance estimate. In the case of the Tully-Fisher relation, the rotational
velocity of a spiral galaxy serves as the distance-independent observable that
is used to predict the total absolute luminosity, which is distance-dependent.
The total luminosity can then be compared with apparent magnitude to measure
distance via the distance modulus. The Tully-Fisher relation has since been
extensively studied and refined Silk1997 ; Avila-Reese1998 ; Kraan-
Korteweg1986 ; O'Neil2000 ; McGaugh2000 ; McGaugh2005 ; Courteau2007 ;
Said2015 . It has become a workhorse tool for measuring the distances and
properties of galaxies, particularly in the context of large-scale structure
surveys and cosmological studies Springob2007 ; Masters2008 ; Tully2009 ;
Courtois2013 ; Hong2019 ; Courtois2023 ; Boubel2023 .

###  0.1.2 Historical Background

The historical background of the Tully-Fisher relation can be traced back to
the Great Debate in the 1920s, when Ernst Öpik used an expression between the
observed rotational velocity and the absolute magnitude to measure the true
distance to Andromeda Opik1922 . This measurement helped to prove that
Andromeda is an independent galaxy, not part of our own Milky Way Galaxy, as
Shapley had thought.

More than 50 years later, Balkowski et al used the Nançay radio telescope in
France to measure the line widths of a sample of 13 irregular and spiral
galaxies Balkowski1974 . They found a correlation between the line width and
the luminosity but they did not apply this correlation as a distance
indicator. This laid the foundation for the ground-breaking paper by Brent
Tully and Richard Fisher in 1977 Tully-Fisher1977 . In their paper, Tully and
Fisher used only inclined spiral galaxies and proposed the usage of the linear
relation between H I profile and absolute magnitude as a distance indicator.

The publication of the Tully-Fisher relation in 1977 and the proposal to use
it as a distance indicator was significant in many different ways. Firstly, it
provided a robust new tool for measuring distance at redshifts that other
methods such as Cepheid variable stars cannot reach. This opened up a whole
range of large scale structure and cosmological studies. Notably, similar
relations like the Fundamental Plane relation Djorgovski1987 ; Dressler1987
only emerged a decade later. Secondly, Tully and Fisher measured the Hubble
constant $H_{0}$ to be 80 $\mathrm{km\ s^{-1}Mpc^{-1}}$ from the Virgo cluster
and Ursa Major. This value was the first to deviate from the two mainstream
values at that time, low value of $H_{0}=50$ $\mathrm{km\ s^{-1}Mpc^{-1}}$
which was promoted by Sandage and Tammann Sandage1975  and a higher value of
$H_{0}=100$ $\mathrm{km\ s^{-1}Mpc^{-1}}$ claimed by de Vaucouleurs
deVaucouleurs1979 . This value is much closer to the value of Hubble constant
that we know today.

###  0.1.3 Tully-Fisher relation in cosmology

The Tully-Fisher relation has been heavily used in measuring the Hubble
constant $H_{0}$. Additionally, it has also been used as a valuable tool in
many other cosmological studies111I do not intend to give a comprehensive
review of all cosmological studies that used TF relation, but I will discuss a
few examples before focusing on the role of TF relation in the $H_{0}$ tension
which is the main focus of this chapter.:

  * •

Cosmography: The Tully-Fisher relation uses the maximum rotational velocity of
a galaxy to predict its luminosity. This can be used to estimate the distance
to the galaxy, and to measure its peculiar velocity. The peculiar velocity is
the deviation from the smooth Hubble flow and can be used to reconstruct 3D
maps of the density and velocity fields of the universe. In 2014, a team led
by Brent Tully used the Wiener filter reconstruction Zaroubi1995  to recover
the 3D density and velocity maps. They used these maps to set the borders of
our home supercluster which they called Laniakea Tully2014 . More recently,
Alexandra Dupuy and Hélène Courtois used the full CosmicFlows-4 Tully2023
catalog222This catalog contains TF distances along with other distance
indicators to reconstruct the 3D density and velocity fields Courtois2023b
and then redefine the boundaries of the Laniakea supercluster along with other
structures such as Perseus-Pisces and Shapley Dupuy2023 .

  * •

Bulk Flow: The bulk flow is the average peculiar velocity over a sphere of a
given radius R. The theoretical expected value of this measurement relies on a
model of a specific set of cosmological parameters, initial conditions and a
chosen gravitational law Andersen2016 . In 2009, Watkins, Feldman and Hudson
used the largest TF survey available at that time, the SFI++ survey
Springob2007 , to measure the bulk flow of galaxies. They used a new method,
the minimal variance estimator, and found a large bulk flow amplitude of
$431\pm 102$ km s-1 within a scale of $50$ Mpc $h^{-1}$ Watkins2009 . The
amplitude of this bulk flow is larger than expected from the $\Lambda$CDM
model. More recently, several teams have used the state-of-the-art
CosmicFlows-4 catalog Tully2023 , which contains more than 10,000 newly
measured TF distances, to measure the bulk flow of galaxies. They have also
found large bulk flows, arising on a much larger scale, that are higher than
the expected value from the $\Lambda$CDM model Courtois2023b ; Watkins2023 ;
Whitford2023

  * •

Growth rate of structure: Over the last three decades, many collaborations
have started to test Einstein’s general theory of relativity over a range of
scales. One way to do that is by measuring the growth rate of cosmic
structure, or $f\\!\sigma_{8}$. This method is capable of measuring
$f\\!\sigma_{8}$ at low redshifts ($z<0.1$), a range inaccessible to other
methods such as the redshift space distortion. The growth rate can be
constrained by comparing the density field from redshift surveys and peculiar
velocity fields from TF surveys. The growth rate can then be parameterized as
a function of the mass density parameter $\Omega_{m}$ and the growth index
$\gamma$, which is determined by the theory of gravity. In 2011, Marc Davis
and others used the TF survey SFI++ to find a $f\\!\sigma_{8}$ value of
$0.31\pm 0.06$ Davis2011 . Other teams also used the TF SFI++ catalog and
found a slightly higher value of $0.401\pm 0.024$ Carrick2015 . More recently,
the CosmicFlows-4 catalog also was used to measure $f\\!\sigma_{8}$ and found
a similar value of $0.40\pm 0.07$ Boubel2023 .

  * •

Galaxy formation and evolution: As an empirical relation between fundamental
properties of galaxies, TF relation has been used to refine semi-analytical
models and numerical simulations of galaxy formation and evolution Navarro2000
; Vogelsberger2014 ; Ponomareva2018 .

  * •

Dark matter in galaxies: The Tully-Fisher relation have also been used to
probe the distribution and properties of dark matter in galaxies. Several
studies put constraints on galaxy halo profiles by comparing the halo profile
required by the Tully-Fisher relation to the density profile that is well-
described by the Navarro-Frenk-White (NFW) profile Navarro1996 ; Mo2000 ;
Seljak2002 .

##  0.2 The Tully-Fisher relation

###  0.2.1 How does it work?

The Tully-Fisher relation is defined and its usages in cosmology are discussed
above. But how does it work in practice?

Building the Tully-Fisher relation requires two data sets: photometry and
spectroscopy. Photometry provides a measurement of the galaxy’s luminosity,
which is the distance-dependent parameter in the TF relation. Spectroscopy
provides a measurement of the galaxy’s rotational velocity, which is the
distance-independent parameter in the relation. The Tully-Fisher relation
establishes the connection between the luminosity and rotational velocity of
spiral galaxies. The TF relation is a secondary distance indicator and as the
name suggests it requires calibration from a sample of galaxies with known
distances usually coming from a primary distance indicator or a sample of
cluster galaxies. Figure 1A represents the initial step in utilizing the TF
relation for distance measurements. It shows a sample of galaxies with known
rotational velocities and absolute magnitudes. The TF relation of the form:

$\displaystyle M=a+b\log W$ (1)

where $M$ is the absolute magnitude, $W$ represents the rotation, $a$ and $b$
are the TF slope and intercept, respectively, can be fit to that sample. This
relation, which we often call a template relation (Fig. 1B), can be used to
predict the absolute magnitude of a galaxy with unknown distance given a
measurement of its rotational velocity as shown in Fig. 1C. The predicted
absolute magnitude can be compared to the galaxy’s apparent magnitude to yield
the distance via the distance modulus as:

$\displaystyle\mu=m-M(W).$ (2)

where $\mu$ is the distance modulus, $m$ is the apparent magnitude, and $M(W)$
is the predicted absolute magnitude of a galaxy from the TF relation given its
rotational velocity

Figure 1: Illustrating how the Tully-Fisher relation combines photometric and
spectroscopic data to enable the estimation of galaxy distance. In order to
use the Tully-Fisher relation to measure distances, we need to build something
called template relation. (A) shows a sample of galaxies with known rotation
velocity, apparent magnitude, and distances which allows the determination of
luminosity. (B) By fitting the TF relation to this sample, a template relation
is derived. This establishes the correlation between the absolute magnitude
and the rotational velocity of galaxies. (C) Applying the Tully-Fisher
template to a galaxy of unknown distance, the absolute magnitude can be
estimated based on its measured rotational velocity.

It is important to note that the derived template relation is not a universal
relation. Instead, the data, parameters, and corrections used in the
measurements of the TF distances should be consistent with those used in
building the TF template relation. This has encouraged many teams to derive
their own template relations instead of using an existing one. In the past few
decades, different measurements, wavelengths, and methodologies have been used
to do this. For example, shortly after the inception of the TF relation,
several studies began to use near-infrared (NIR) bands because they suffer
less than optical bands from dust extinction (both internal and external) and
are more sensitive to stellar mass Aaronson1979 ; Masters2008 ; Said2015 .
Other groups used isophotal magnitudes instead of total magnitudes to extend
the Tully-Fisher studies into the Zone of Avoidance around the plane of the
Milky Way Bouche2000 ; Said2016b .

In terms of spectroscopy, many parameters have been used in an attempt to
minimise the intrinsic scatter of the TF relation. Many teams have used the
20% H I line width (the width measured at 20% of the peak flux of the H I
profile) including the original TF relation Tully-Fisher1977 ; Pierce1988 ;
Courtois2009 , while others suggested using the width measured at 50% instead
Giovanelli1997 ; Masters2006 ; Said2016a ; Bell2023 . Furthermore, other teams
suggest that the flat part of the optical rotation curve is the best proxy for
the rotational velocity, and they have shown that it does indeed yield the
tightest Tully-Fisher relation Verheijen2001 ; Lelli2019 .

Different methodologies have been used to extract the best-fitting parameters
for the Tully-Fisher relation by applying different fitting procedures. In an
attempt to overcome the Malmquist bias, Renée C. Kraan-Korteweg and others
proposed the use of the inverse TF relation, which only takes into account the
errors on the x-axis (rotational velocity) Kraan-Korteweg1986 . Since then,
many teams have used both direct and inverse fitting procedures in their
analyses Willick1995 ; Willick1996 . Other teams have used bivariate forms
that take into account errors in both the x- and y-axes Giovanelli1997 ;
Masters2008 ; Said2015 ; Bell2023 . Given the scatter in TF relation, a
suggestion has been made to use a Bayesian mixture model, which is less biased
by outliers Hogg2010 .

In summary, the Tully-Fisher relation has been observed to hold over a wide
range of applications, including different parameters, methodologies, and
wavelengths. This suggests that the TF relation is one of the most robust and
powerful tools for measuring galaxy distances.

###  0.2.2 Theoretical basis

The physical origin of the Tully-Fisher relation is still not fully
understood, but it is certainly rooted in the physics of gravity and the
dynamics of galactic rotation. The TF relation links the mass of a galaxy
(characterized by the luminosity) to its dynamics (characterized by the
rotational velocity). Specifically, it is based on the idea that the rotation
velocity of a galaxy is related to the mass contained within a certain radius
from the galaxy’s center, known as the circular velocity.

Michael Strauss and Jeffrey Willick provided a widely accepted theoretical
explanation333This was not the first such explanation, but it was particularly
clear. for the Tully-Fisher relation in their 1995 review Strauss1995 . They
wrote that by equating the centrifugal force of an object moving in a circle
of radius r to the gravitational attraction on the same object due to the mass
inside a sphere of radius r, one can write

$\displaystyle V_{\text{rot}}^{2}\propto\frac{M}{r}$ (3)

where $V_{\text{rot}}$ is the rotational velocity, $M$ is the mass within a
sphere of radius $r$, and $r$ is the distance from the galaxy’s center. Then,
assuming a constant value for both the mass-to-light ratio (M/L) and mean
surface brightness for spirals, one can obtain that

$\displaystyle L\propto V_{\text{rot}}^{4}$ (4)

However, most Tully-Fisher studies have derived a power-law exponent that
deviates from this theoretical explanation. This is usually attributed to the
complication of the dark matter halo that surrounds a galaxy Mould2020 . Early
near-infrared Tully-Fisher analyses showed that the complications of halo mass
could be avoided, and indeed derived a power-law exponent that was closer to
the theoretical expectation Aaronson1979 . However, their conclusion has been
contradicted by many larger subsequent surveys Giovanelli1997 ; Masters2008 .

In 2000, Stacy McGaugh and his team investigated the deviation of dwarf faint
galaxies from the linear Tully-Fisher relation McGaugh2000 . They found that
the linear TF relation could be restored if they used the sum of both stellar
and gas mass components instead of luminosity. This relation takes the form

$\displaystyle M_{d}\propto V_{\text{rot}}^{4}$ (5)

where $M_{d}$ is the sum of stellar and gas masses,

$\displaystyle M_{d}=M_{*}+M_{\text{gas}}.$ (6)

Not only did they establish the baryonic TF relation as the fundamental
relation, but they also found that the intrinsic scatter was smaller than the
original TF relation. This result can be simply explained if we suppose that
this estimate of the baryonic mass is a better proxy for the true total mass
than the luminosity or the stellar mass alone.

This result, at first glance, may seem to support the Modified Newtonian
Dynamics (MOND: Milgrom1983 ) as an alternative to dark matter. MOND is a
theory of gravity that modifies Newtonian dynamics. In MOND, the baryonic mass
is actually the total mass, which means that the baryonic TF relation is the
fundamental TF relation. However, it is important to note that Modified
Newtonian Dynamics (MOND) has its own critics vandenBosch2000 ;
vandenBosch2001 . We refer the reader to the literature for more information
about the debate between MOND and Cold Dark Matter.

One important theoretical development in recent years has been the use of
hydrodynamic simulations to study the formation and evolution of galaxies in a
cosmological context Vogelsberger2014 ; Schaye2015 . These simulations
incorporate the effects of gravity, gas dynamics, star formation, and
feedback, and have been shown to predict the Tully-Fisher relation in
different environments and at different redshifts better than semi-analytical
models Papastergis2016 .

###  0.2.3 Advantages and limitations

The Tully-Fisher relation has several advantages as a tool for measuring
distances of galaxies. The first and main advantage is its simplicity and
robustness. The TF relation provides a linear correlation between only two
easily measurable galaxy properties, luminosity and rotational velocity.
Several studies have shown that, unlike other distance indicators, the scatter
of the TF relation is relatively insensitive to other galaxy properties
Courteau1999 ; Courteau2007 ; Ouellette2017 . This makes the TF relation a
fundamental, robust and reliable tool for measuring distances.

Another advantage of the Tully-Fisher relation is that it uses spiral
galaxies, which are the most numerous type of galaxies at low redshifts
($z<0.07$). This makes it a convenient tool for measuring peculiar velocities
at these redshifts, where the true distances can be determined. Beyond this
redshift, it is more difficult to obtain accurate peculiar velocities as the
errors in measuring these velocities from distance indicators scale with
redshift.

[scale=.465]s_e_sn.png

Figure 2: The redshift distribution of spiral (black) and elliptical (red)
galaxies in the SDSS survey, as well as the redshift distribution of the
state-of-the-art SN Ia from Pantheon+ (blue), up to redshift $z<0.15$. The
number of galaxies per 0.0025 redshift bin is shown.

In Figure 2, I show the redshift distribution of spiral and elliptical
galaxies from the SDSS survey DR14 Abolfathi2018 . I separated spirals from
ellipticals using the galaxy zoo catalog Lintott2011 . I also over plot the
redshift distribution of the state-of-the-art Pantheon+ sample Carr2022 ;
Brout2022 . I used only the redshift below $z<0.15$, as this is the range used
for Hubble constant ($H_{0}$) measurements Riess2016 . It is clear that spiral
galaxies (Tully-Fisher galaxies) are the most numerous type at low redshift
$(z<0.07)$, with a mean redshift of $z=0.077$. This makes TF relation one of
the most precise tools for measuring $H_{0}$ at this redshift.

However, there are also some limitations to the Tully-Fisher relation as a
tool for distance measurements. One of the main limitations of the TF relation
is its intrinsic scatter. This scatter can lead to errors of around $\sim$20%
in distance measurements. The baryonic Tully-Fisher relation is often shown to
be a tighter relation than the TF relation, with a much smaller intrinsic
scatter McGaugh2012 ; Lelli2016 . However, measuring the sum of the stellar
and gas components is much more difficult than measuring just the luminosity
of a galaxy. This adds a lot of observational uncertainties to the baryonic TF
relation.

Another limitation of the Tully-Fisher relation and a significant source of
uncertainty is the inclination of galaxies. The inclination of a spiral galaxy
affects both TF parameters. It affects the derived maximum rotational velocity
during the process of correcting the observed line width for projection
effects (i.e. to edge-on orientation). Additionally, to correct the magnitude
for the internal extinction, one needs to use the inclination to account for
the path-length of the light through the galaxy. Therefore, inclination errors
lead to correlated errors in the two TF parameters that have the same sense
(qualitatively) as errors in distance, so an error in inclination is (at least
partly) degenerate with an error in distance. Most Tully-Fisher surveys select
only edge-on galaxies by applying a cut to inclinations in an attempt to
minimize the line width correction. Although this seems like the right thing
to do, inaccurate inclinations will lead to a selection bias in the TF
distance. This is because galaxies with inaccurate inclinations may be
excluded or included in the survey, depending on whether their inclinations
are overestimated or underestimated. Furthermore, edge-on galaxies are subject
to higher internal extinction, which means that more light is blocked by dust
and gas inside the galaxy. This results in a dimmer magnitude, and thus a
larger correction is needed to correct for the extinction. Some studies have
even suggested that when the inclination measurements have errors larger than
10 degrees, it may be more prudent to omit inclinations entirely rather than
assuming them to be exact Obreschkow2013 .

Selection effects and Malmquist bias are other limitations of the TF relation.
However, these limitations are not specific to the TF relation, as they affect
all distance indicators.

##  0.3 The Role of the Tully-Fisher relation in $H_{0}$ Measurements

The methodology for using the Tully-Fisher relation in $H_{0}$ measurements
typically involves the following steps. First, a sample of galaxies with well-
measured rotational velocities and luminosities (for TFR) or masses (for the
BTFR) is selected (see Fig. 1A). These galaxies are typically chosen to be
representative of the population of galaxies being studied. Next, the Tully-
Fisher relation is calibrated for this sample of galaxies. This involves
measuring the slope, intercept, and scatter of the relation for the sample,
and correcting for any systematic biases or selection effects (see Fig. 1B).
Once the Tully-Fisher relation has been calibrated for the sample, it can be
used to infer the distances to other galaxies with similar properties (see
Fig. 1C). The final step is measuring $H_{0}$, which can be done by plotting
the derived TF distances against redshifts.

This methodology has been used in a number of studies to estimate the value of
the Hubble constant ($H_{0}$). In 1977, Tully and Fisher applied their newly
discovered Tully-Fisher relation to derive distances to the Virgo and Ursa
Major clusters. They measured a Hubble constant of $H_{0}=84$ km s-1 Mpc-1 for
Virgo and $H_{0}=75$ km s-1 Mpc-1 for Ursa Major. They conclude a preliminary
value for the Hubble constant of $H_{0}=80$ km s-1 Mpc-1. Although they called
it preliminary and did not propagate their distance error to the Hubble
constant value, we know today that it was the most accurate result at that
time.

In 1983, Visvanathan used the Tully-Fisher relation to measure distances to 52
cluster spiral galaxies. They found that the Hubble constant derived from
nearby clusters varied from 58.5 to 83.5, but the range was much smaller for
distant clusters, between 76.3 and 78.9 km s-1 Mpc-1. Visvanathan concluded
that the best fit value for the Hubble constant from all clusters in their
study was $H_{0}=74.4\pm 11$ km s-1 Mpc-1 Visvanathan1983 . That result is
very close to the value of $H_{0}$ we have today.

Shortly after, Sandage and Tammann used the Tully-Fisher relation in the
infrared (IR) and blue bands to estimate distances to galaxies in the Virgo
and Coma clusters, respectively. Their analysis gave a lower value of the
Hubble constant $H_{0}=55\pm 9$ km s-1 Mpc-1 Sandage1984 . This result aligned
with their previous measurement of $H_{0}=50\pm 7$ km s-1 Mpc-1 obtained
through Type I supernovae Sandage1982 .

Aaronson led a team in 1986 to derive distances to 10 galaxy clusters using
the infrared Tully-Fisher relation. They stated that the current evidence
favors a large value for $H_{0}$ of 90 km s-1 Mpc-1 Aaronson1986 . In two
subsequent studies, Bottinelli led a team to use the infrared Tully-Fisher
relation to derive distances to galaxy clusters. They concluded that the
$H_{0}$ lies between 70 and 75 km s-1 Mpc-1. During the same time, Renée C.
Kraan-Korteweg and others derived distances to a larger sample of 82 cluster
galaxies using both the infrared Tully-Fisher relation and the optical Tully-
Fisher relation. They found that their data is consistent with a Hubble
constant of $H_{0}=56.6\pm 0.9$ km s-1 Mpc-1 Kraan-Korteweg1988 . However,
this low value was contradicted by three studies. One study used the TF
relation to measure distances to Virgo and Ursa Major clusters, and derived a
value of $H_{0}=85\pm 10$ km s-1 Mpc-1 Pierce1988 . The other two studies
derived distances to the Coma cluster, and also found a higher value of
$H_{0}\approx 90$ km s-1 Mpc-1 Fukugita1991 ; Rood1993 . Bureau, Mould, and
Staveley-Smith Bureau1996  used the Tully-Fisher relation to measure distances
to galaxies in the Fornax cluster, and derived a value of $H_{0}=74\pm 11$ km
s-1 Mpc-1.

This significant discrepancy in these early measurements can be attributed to
various factors, including the challenge of sample incompleteness correction
and the absence of reliable distances to nearby objects for use as a
calibration sample.

The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
used Cepheid distances to re-calibrate the Tully-Fisher relation. In a series
of papers, they found results consistent with a value of $H_{0}$ of $71\pm 4$
km s-1 Mpc-1 Madore1999 ; Sakai2000 ; Freedman2001 .

The Cosmicflows project combines distances measured through various methods,
including the Tully-Fisher relation, surveys, and collaborations. This project
has made significant contributions to measuring the Hubble constant,
consistently obtaining values in the range of $74\pm 4$ to $76\pm 1$ km s-1
Mpc-1 over an extended period Tully2012 ; Sorce2012 ; Sorce2013 ; Neill2014 ;
Sorce2014 ; Kourkchi2020 ; Kourkchi2022 ; Courtois2023b

Figure 3 shows a graphical representation of the published efforts to measure
the Hubble constant ($H_{0}$) using the Tully-Fisher relation. The red band
shows the the most recent result of Hubble constant and its associated errors
measured using Type Ia supernovae by the SH0ES team in 2022 Riess2022 .
Additionally, the blue band represent the Hubble constant value and its
associated error from the CMB measured by the Planck Collaboration in 2020
PlanckCollaboration2020 . The figure shows a lot of scatter in the TF results
prior to 2000 with some unrealistically optimistic uncertainties. However,
after 2000, the results seem consistent and in more agreement with the value
derived by the SH0ES team in 2022 Riess2022  than the CMB value by the Planck
Collaboration in 2020 PlanckCollaboration2020 . Table 0.3 provides a summary
of the results from these studies.

Figure 3: A sample of published Hubble constant ($H_{0}$) measurements using
Tully-Fisher relation spanning from 1977 to 2023. Most of these measurements
are obtained by different teams using different data given in table 0.3. The
red band highlights the most recent Hubble constant value and its associated
uncertainty from the baseline results of the Cepheid-SN Ia sample by the SH0ES
team Riess2022 . The blue band represents the Hubble constant value reported
by Planck Collaboration 2020 PlanckCollaboration2020 . The majority of the
data points align closely with the red band, particularly from 2000 onward. It
is important to note that this sample of data is just a subset from the
literature and additional studies may already exist beyond those represented
here. Table 1: Evolution of published Hubble constant values from the Tully-
Fisher relation (1977-2023)

Date |  $H_{0}$ |  Ref. |  |  Date |  $H_{0}$ |  Ref.
---|---|---|---|---|---|---
1977 |  80.0 |  Tully-Fisher1977  |  |  2001 |  71.0 $\pm$ 3.0 |  Freedman2001
1983 |  74.3 $\pm$ 11.0 |  Visvanathan1983  |  |  2001 |  73.0 $\pm$ 4.0 |  Watanabe2001
1984 |  55.0 $\pm$ 9.0 |  Sandage1984  |  |  2006 |  74.0 $\pm$ 2.0 |  Masters2006
1986 |  90.0 |  Aaronson1986  |  |  2009 |  84.2 $\pm$ 6.0 |  Russell2009
1987 |  72.0 $\pm$ 5.0 |  Bottinelli1987  |  |  2011 |  79.0 $\pm$ 2.0 |  Hislop2011
1988 |  85.0 $\pm$ 10.0 |  Pierce1988  |  |  2012 |  75.0 |  Tully2012
1988 |  72.5 |  Bottinelli1988  |  |  2012 |  75.2 $\pm$ 3.0 |  Sorce2012
1988 |  56.6 $\pm$ 0.9 |  Kraan-Korteweg1988  |  |  2013 |  74.0 $\pm$ 4.0 |  Sorce2013
1991 |  92.0 $\pm$ 21.0 |  Fukugita1991  |  |  2014 |  74.4 $\pm$ 1.0 |  Neill2014
1993 |  90.0 |  Rood1993  |  |  2014 |  75.2 $\pm$ 3.3 |  Sorce2014
1996 |  74.0 $\pm$ 11.0 |  Bureau1996  |  |  2020 |  75.1 $\pm$ 2.3 |  Schombert2020
1999 |  76.0 $\pm$ 3.0 |  Madore1999  |  |  2020 |  76.0 $\pm$ 1.1 |  Kourkchi2020
2000 |  77.0 $\pm$ 8.0 |  Tully2000  |  |  2022 |  75.5 $\pm$ 2.5 |  Kourkchi2022
2000 |  71.0 $\pm$ 4.0 |  Sakai2000  |  |  2023 |  74.5 $\pm$ 0.1 |  Courtois2023b

a This is just a subset of the published Hubble constant values using TF
relation, and additional studies may already exist beyond those represented
here.

In summary, the Tully-Fisher relation has been used for decades to estimate
distances to galaxies and to measure the Hubble constant. Although Tully-
Fisher measurements of $H_{0}$ have shown a lot of scatter prior to 2000, they
have been consistent since then with a value of $H_{0}$ that agrees with the
other low-redshift probes.

##  0.4 Future Prospects

The largest homogeneous Tully-Fisher sample currently exists is the
CosmicFlows-4 survey Kourkchi2020 . This survey includes 10,000 TF distances.
Imminent surveys such as DESI Saulder2023  and WALLABY Courtois2023  will
provide 50,000 and 200,000 TF distances, respectively, over the coming few
years. With approximately 20% TF distance errors for about 250,000 galaxies
covering a significant portion of the sky up to redshift $z\approx 0.1$, both
random and systematic errors in $H_{0}$ from large-scale structure will be
considerably smaller compared to the most optimistic uncertainties associated
with other low-redshift probes.

###### Acknowledgements.

KS acknowledges support from the Australian Government through the Australian
Research Council’s Laureate Fellowship funding scheme (project FL180100168). I
would like to express my gratitude to Tamara Davis and Matthew Colless for
their valuable feedback on the initial draft of this review.

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