In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $k$ firefighters protect $k$ vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect $k$ vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. A common objective with respect to some infinite graph $G$ is to determine how many firefighters are necessary to stop the fire from spreading after a finite number of turns, commonly referred to as containing the fire. We introduce the concept of distance-restricted firefighting where the firefighters' movement is restricted so they can only move up to some fixed distance $d$ per turn rather than being able to move without restriction. We establish some general properties of this new game in contrast to properties of the original game, and we investigate specific cases of the distance-restricted game on the infinite square, strong, and hexagonal grids. We conjecture that two firefighters are insufficient on the square grid when $d=2$, and we pose some questions about how many firefighters are required in general when $d=1$.

§ INTRODUCTION

In this paper we introduce the concept of the distance-restricted firefighter game based on Hartnell's firefighter game introduced in [7]. The original game is a turn-based game played on a graph where fire is a proxy for some contagion that spreads from a set of `burning' vertices to their collective neighbourhoods on each turn. For our purposes we consider the game to be a two player game where the players are an arsonist and a brigade of firefighters. On turn $0$, one or more vertices are selected by the arsonist as the initial burning vertices and some fixed number of firefighters are placed as well. Each firefighter protects a single vertex from the fire, the consequence being that the fire does not spread to these vertices for the remainder of the game. On each subsequent turn, the fire spreads to the unprotected neighbourhoods of the burning vertices and the firefighters protect a new set of vertices. The firefighters are generally trying to achieve a predetermined goal, which in our case will be to reach a point where the fire can no longer spread. This goal is referred to as containing the fire and is most relevant to infinite graphs, which will be our focus. For a survey of results on firefighting, we recommend the paper by Finbow and MacGillivray [4] or the recent thesis by Wagner [10].

In the original game there is no relationship between the firefighters' positions at time $t$ and time $t+1$, which is not always a good representation of the real world since in reality, firefighters are restricted by how fast they can move. Our variation of the rules, which we have dubbed distance-restricted or distance $d$ firefighting, seeks to make the game more realistic by forcing each firefighter to only move a maximum of some specified distance $d$ between time $t$ and time $t+1$. This condition can be explicitly defined by saying that given a fixed, positive integer $d$ and $k$ firefighters labelled as $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_k$, then for every $t\in\mathbb{N}$, there must exist a bijection $g_t$ between the sets POS$_t$ and POS$_{t+1}$ where POS$_t = \{(\mathcal{F}_i,u_i) \mid \mathcal{F}_i \text{ occupies vertex }u_i\text{ at time }t\}$ and $g_t(\mathcal{F}_i,u) = (\mathcal{F}_j,v)$ implies that the distance between $u$ and $v$ is at most $d$ and that $i=j$. We can see here that there is an ambiguity in whether or not the distance between $u$ and $v$ is determined with respect to the entire graph or just the subgraph induced by the unburnt vertices. We will be using the distance in the induced subgraph since it is a more realistic model, but we will also occasionally make note of how things may be different if we allow the firefighters to move through the fire. We will also note that our definition of the distance-restricted game requires some refinement to make it compatible with instances where the number of firefighters available varies from turn to turn. If the number of firefighters increases by $\ell$ on any given turn, then before the firefighters move, the $\ell$ additional firefighters are placed on vertices that currently have at least one firefighter on them, and then the firefighters move as normal. If the number of firefighters decreases by $\ell$ on any given turn, then before the firefighters move, $\ell$ of them are chosen and removed from play, and then the remaining firefighters move as normal.

Some notation is now introduced for defining how many firefighters are required to contain a fire on a graph. In the context of the original game, we define $f(G,u)$ to be the minimum number of firefighters needed to contain a fire that breaks out at the vertex $u$ in the graph $G$ and we refer to this quantity as the firefighter number of $G$ for a fire at $u$. If the graph is vertex transitive, like the grids we will be focusing on, we can simply write $f(G)$ since the choice of $u$ does not matter and we refer to this simply as the firefighter number of $G$. We can then analogously define $f_d(G,u)$ by considering the distance $d$ firefighting game and refer to $f_d(G,u)$ and $f_d(G)$ as the distance-restricted firefighter number. The notation $f^*_d(G,u)$ is defined analogously to $f_d(G,u)$, except that the firefighters are allowed to move through the fire.

In the following sections, we first discuss some general properties of distance-restricted firefighting in contrast to the original game. We then continue by determining the value of $f_d(G)$ for various values of $d$ on the square, strong, and hexagonal grids. Special attention is given to the square grid as the case when $d=2$ is of particular interest. We conjecture that in this case, two firefighters are insufficient to contain the fire. In Section <ref> we outline a special case of a conjecture by Messinger [8] and give a brief overview of how one might attempt to prove this special case and why it is likely easier to prove than Messinger's conjecture. In the final section we pose some questions about some general properties of the game as well as some questions about the game when $d=1$ specifically.

§ GENERAL PROPERTIES OF DISTANCE-RESTRICTED FIREFIGHTING

There are some nice properties of the original game that the distance-restricted game unfortunately does not have. For example, in the original game, if $H$ is a subgraph of $G$ with $u \in V(H)$ then $f(G,u) \geq f(H,u)$. This is apparent from the fact that a strategy that contains the fire with $k$ firefighters on $G$ can be slightly modified and then played on $H$ as well. For any placement of the firefighters, there are two cases to consider. At time $t$, if all the vertices that would have been protected in the strategy on $G$ exist in $H$, then the firefighters protect those vertices. Otherwise, if they do not all exist in $H$ then any missing vertices can be considered as protected since the fire cannot spread to them and so rather than protecting those missing vertices, an arbitrary set of vertices can be protected without hindering the strategy. Since the fire cannot spread to any additional vertices due to the fact that all the same vertices from $G$ are either protected or missing, the fire cannot spread in $H$ more than it did on $G$ and thus the fire is contained in $H$ as it is contained in $G$. However, this property is not true in the distance-restricted firefighting game. This fact will become apparent shortly after observing some properties of certain infinite grids.

As noted in [9] the infinite square ($G_{\square}$), triangular ($G_{\triangle}$) and strong ($G_{\boxtimes}$) grids have a relationship such that $V(G_{\square}) = V(G_{\triangle}) = V(G_{\boxtimes})$ and $E(G_{\square}) \subset E(G_{\triangle}) \subset E(G_{\boxtimes})$. This nesting can be further extended to the infinite hexagonal ($G_{\hexagon}$) grid by noting $V(G_{\hexagon}) = V(G_{\square})$ and $E(G_{\hexagon}) \subset E(G_{\square})$ as illustrated in Figure <ref>. Clearly the illustrated grid is contained in the square grid, and it is easy to observe that the grid in Figure <ref> is isomorphic to the hexagonal grid.

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$7 \times 7$ Portion of the hexagonal grid as a subgraph of the square grid

Observe in Figure <ref> that the hexagonal grid contains a subdivision of itself. The paths of length $3$ formed by the degree $2$ vertices and their neighbours can be replaced with single edges in order to recover the standard hexagonal grid.

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$7 \times 7$ Portion of the subdivided hexagonal grid as a subgraph of the hexagonal grid

The subdivided hexagonal grid (sub($G_{\hexagon}$)) only requires one firefighter when $d \geq 11$ as depicted[There are many diagrams in this article that all follow the same labelling conventions, which we outline here. The firefighters are usually labelled so that the even numbered turns have black vertices and the odd numbered turns have gray vertices. The fire follows a similar convention with the even turns being red and the odd turns being orange. We also adopt the convention that the first turn is turn zero and the vertices where the fire and firefighters start are diamond shaped. There are also diagrams where there is no fire, and in those cases the first firefighters may be diamond shaped if necessary for clarity. We also occasionally have a firefighter who is distinguished from the other firefighters in some way, and in those cases the firefighter may be coloured blue instead of black and gray.] in Figure <ref>, but sub($G_{\hexagon}$) contains an infinite path[Simply consider any set of vertices $\{(x,y)|x \in \mathbb{Z}\}$ for some fixed value of $y$.] which stretches out in two directions. This subgraph requires two firefighters to contain the fire for any value of $d$. Thus the firefighter number of a subgraph is not bounded above by the firefighter number of the original graph in the distance-restricted game. In fact Figure <ref> and Theorem <ref> show that the ratio and difference of $f_d(G)$ and $f_d(H)$ for $H$ a subgraph of $G$ can both be unbounded.

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A strategy to contain the fire in $4$ turns on the subdivided hexagonal grid when $d=11$. The case where the fire begins on a degree $2$ vertex is trivial and thus omitted.

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Left: The first $3$ layers of $P_{0,\infty}\:\square\:C_5$ modified as described in the proof of Theorem <ref>, Right: An infinite subgraph of the graph on the left with distance-restricted firefighter number $5$

There exist graphs $G,H$ such that $H$ is a subgraph of $G$ with $u \in V(H) \subseteq V(G)$ such that the value of $\big(f_d(H,u) - f_d(G,u)\big)$ as well as the value of $\Big(\frac{f_d(H,u)}{f_d(G,u)}\Big)$ can be arbitrarily large for any value of $d$.

The graph in Figure <ref> can be thought of as taking $P_{0,\infty}\:\square\:C_m$ (the Cartesian product of an infinite ray and a cycle) and replacing the cycle corresponding to the end vertex of the ray with a wheel on $m+1$ vertices. This graph has distance-restricted firefighter number $1$ for a fire breaking out at the central vertex for any value of $d$ since if we start our firefighter in the $(m+1)^{th}$ cycle then the firefighter can just walk around the cycle and form a barrier. However if we consider the subgraph depicted in Figure <ref>, then any fire breaking out at the central vertex will require $m$ firefighters to contain the fire no matter the value of $d$. Thus both the difference and the ratio are unbounded for any value of $d$.

This leads us to the question of how large the difference between the firefighter number and the distance-restricted firefighter number can be.

The value of $\big(f_d(G,u) - f(G,u)\big)$ as well as the value of $\Big(\frac{f_d(G,u)}{f(G,u)}\Big)$ can be arbitrarily large for any value of $d$.

Simply observe the subgraph from the proof of Theorem <ref> as depicted in Figure <ref> (right). This graph clearly only requires $1$ firefighter in the original game, but requires $m$ firefighters in the distance-restricted game. Thus both the ratio and the difference are unbounded here.

Observe that Theorem <ref> does not extend to when the firefighters are allowed to pass through the fire (i.e. it is false that $\big(f^*_d(G,u) - f(G,u)\big)$ can be arbitrarily large for any value of $d$). If $G$ is a connected graph where $f$ firefighters suffice in the original game, then there is a finite distance $d$ where for all $t$ the firefighters at time $t$ and the firefighters at time $t+1$ are within distance $d$. The existence of $d$ is apparent from the fact that the firefighters only move for a finite number of turns so it is guaranteed that a maximum distance moved by the firefighters exists. This is due to the fact that the firefighters can move through the fire so there is always a set of paths from the positions at time $t$ to the positions at time $t+1$. Thus there is always a value of $d$ where the values of $f(G,u)$ and $f^*_\delta(G,u)$ are the same for all $\delta \geq d$.

Observe as well that something similar can be said of Theorem <ref>. Consider $f^*_d(G,u)$ and $f^*_d(H,u)$ for $H$ a subgraph of $G$. There will exist a value of $d$ where $f^*_d(G,u) = f(G,u)$ and $f^*_d(H,u) = f(H,u)$, and so any properties of their ratio and difference are the same as in the original game.

§ SQUARE GRID

Our consideration of $G_{\square}$ can be divided into two parts: the case of $d=2$ and the case of $d\neq2$. The case of $d=2$ has shown to be much more complex than the case of $d\neq2$ so the majority of this section focuses on this more complex case.

§.§ Square Grid when $\mathbf{d\neq2}$

The case of $d=1$ on the square grid was shown to require four firefighters in [1, 2]. The idea in this case is that in order to stop the fire from burning along the four infinite straight paths that start at the origin, there must be four firefighters, and four firefighters are obviously sufficient because the fire can be trivially surrounded on turn zero.

The case of $d=3$ also has an easy solution. Figure <ref> portrays a strategy that works with two firefighters. The two firefighters form a barrier by spiraling around the fire until they are able to contain the fire.

Two firefighters are necessary and sufficient to contain the fire on the square grid in the game with $d \geq 3$.

Figure <ref> illustrates a strategy with two firefighters when $d=3$. Thus two firefighters suffice when $d=3$. For any distance greater than three the same strategy can be played. Thus for any $d \geq 3$ two firefighters suffice.

One firefighter does not suffice since if one firefighter was sufficient for $d=3$ then one firefighter would be sufficient for the original game which is not true from [11].

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A strategy for containment when $d=3$. For extra clarity, this figure is augmented so that the turn numbers are visible on the vertices.

§.§ Square Grid when $\mathbf{d=2}$

As previously mentioned, the case of $d=2$ is much more challenging than the case of $d\neq2$, so we need to introduce more complex strategies. In Theorem <ref> we refer to corralling the fire to a column, which simply means that the two firefighters are moving in the same direction in two parallel lines and if they continued like this then only vertices between those two lines would burn (See Figure <ref>). We also require that this column of fire is only burning in one direction, rather than in two directions.

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Two firefighters corralling the fire to a column

Let $d=2$. If two firefighters can corral the fire to a column of finite width in the infinite square grid, then they can contain the fire.

The proof of this theorem relies on the fact that two firefighters can contain any finite source fire in the half grid when $d=2$. To see this consider a half diamond of fire along the border of the half grid such that the diamond of fire extends out some distance $k$ from the vertices on the border of the grid (see Figure <ref>). Initially, place the firefighters at the edge of the fire with one firefighter on the border and the other on the first firefighter's neighbour that is not on the border (see Figure <ref>). From here the firefighters move upwards until they are $2k$ vertices above the peak of the fire, and they then move over the fire and are high enough above the fire that they will have a clear path across. They can then follow the same strategy but rotated ninety degrees to get far enough past the far edge of the fire that they can then move downwards and contain the fire. Since this initial diamond can be arbitrarily large it can cover all sources of fire, and thus any finite source fire can be contained as well.

So now suppose the fire is contained to a column in the full grid. Both firefighters can continue building the protective barriers on either side of the column for an arbitrarily large number of turns, thus these protective barriers can be arbitrarily long. One of the firefighters continues to move up the side of the fire while the other begins to move two spaces at a time in order to move up above the fire, across the top, and down to where the other firefighter is. At this point the firefighters can build a protective barrier up ahead of the fire on that side and then build a protective barrier across so they are now back in the column where the firefighter that was skipping vertices was. From here observe that there is a finite number of vertices burning on the outer side of the protective barriers that form the column, and the length of these protective barriers is arbitrarily large. Thus the firefighters can contain the fire if they treat the remainder of the game as if the protective barrier of the column is the border of a half grid since both firefighters are on the same side of that border as the fire.

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The beginning of the strategy described in the proof of Theorem <ref> with $k=3$. Note that this is a half grid for which the border is along the bottom of the diagram.

We conjecture that the converse of Theorem <ref> also holds.

Let $d=2$. Then two firefighters can contain the fire in the square grid if and only if they can contain the fire to a column.

Determining $f_2(G_{\square})$ has proven to be a very stubborn problem. We can easily establish that $f_2(G_{\square}) \geq 2$ since two firefighters are optimal in the original game from [11]. With three firefighters the fire is contained in seven turns as illustrated in Figure <ref>, and thus $2 \leq f_2(G_{\square}) \leq 3$.

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A strategy for containment with three firefighters when $d=2$

However, a successful strategy with two firefighters remains elusive. It appears that two firefighters can only contain the fire to a quarter plane in the case of $d=2$. On the other hand, observe that the square grid without the vertices of the form $(0,n)$, for all $n\in\mathbb{Z}^{-}$, is containable with two firefighters when $d=2$.

Let $d=2$. A fire in the infinite square grid, minus the vertices of the form $(0,t)$ for all $t \in \mathbb{Z}^{-}$, can be contained by two firefighters.

First, if the fire starts on a vertex $(\pm 1,t)$ for some $t \in \mathbb{Z}^-$ then the first firefighter starts directly above the fire and the second starts right of the fire if the fire is at $(1,t)$ and left of the fire if the fire is at $(-1,t)$. The firefighters then build a wall downwards through $(\pm 2,t)$ until they are ahead of the fire which is guaranteed because the firefighters are building the wall two vertices at a time and the fire is moving one vertex at a time. Then the firefighters turn in towards the $y$-axis and have thus surrounded the fire.

Now suppose the fire starts on a vertex that is not of the form $(\pm 1, t)$. Then the firefighters can corral the fire to a quarter plane within the modified square grid. This quarter grid can always be set up so that one firefighter ($\mathcal{F}_1$) is moving in the negative $y$-direction and the other ($\mathcal{F}_2$) is moving away from the missing vertices in the $x$-direction. If $\mathcal{F}_1$ is between the fire and the missing vertices, then $\mathcal{F}_1$ turns and moves towards the missing vertices until reaching the missing vertices. Otherwise it must be that $\mathcal{F}_1$ is on the line $x=0$ and so will continue moving in the negative $y$-direction until reaching the missing vertices. From here it is easy to observe that $\mathcal{F}_1$ can retrace steps and then follow the steps of $\mathcal{F}_2$ at twice the speed as before and meet up with $\mathcal{F}_2$. At this point the firefighters can clearly contain the fire.

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The beginning of the firefighters' strategy for the grid with a straight path removed

It is difficult to remove a simpler infinite connected structure from the grid, so we can see that containing the fire here with two firefighters is very close to being able to contain the fire in the original grid. Based on this it appears that two firefighters are nearly sufficient in the square grid when $d=2$. This notion of being nearly sufficient is further strengthened by Lemma <ref>.

In the firefighter game with two firefighters on the infinite square grid with $d=2$, if there is one extra firefighter on an unspecified turn, then the firefighters can contain the fire.

Without loss of generality, suppose the fire starts at $(1,1)$ since $G_{\square}$ is vertex transitive. Observe that if the fire is initially corralled to a quarter plane by moving along the positive $x$- and $y$-axes, then when the extra firefighter is available one of the firefighters can turn ninety degrees and corral the fire to a column. Theorem <ref> then tells us the fire can be contained.

Figure <ref> depicts examples of how some strategies work when deploying the extra firefighter on different turns. The fire starts in the bottom left of the column in all four diagrams on the left with the extra firefighter being deployed on the fourth, third, second, and first turns respectively. Figure <ref> (Left) shows how the right firefighter moves over to meet the left firefighter, and Figure <ref> (Right) shows an example of how the two firefighters subsequently move over to the top of the fire.

Some examples of the strategy with 2,2,…,2,3,2,2,…firefighters. The right image is zoomed in to make the extra firefighter (blue) more visible

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(Left) How the right firefighter moves to join the left firefighter. (Right) How the two firefighters move to be above the fire

We can also allow for a slightly different variation on our rules to further solidify that two firefighters are almost sufficient in the case of $d=2$. If we require that the firefighters' average distance moved on each turn is at most two, then the firefighters can very nearly contain the fire. In fact the firefighters can contain the fire if they are permitted to move through the fire once. We refer to this modified game as sum-distance firefighting and we can see the aforementioned containment with two firefighters in Figure <ref>. Observe that the firefighters have the fire corralled to a column at the last turn in the figure, so by Theorem <ref> the firefighters can contain the fire.

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The initial part of the strategy for containing the fire in the sum-distance game with two firefighters when $d=2$

From Theorem <ref>, Lemma <ref>, Lemma <ref>, and Figure <ref> it is clear that very slight deviations from the problem of containing the fire with two firefighters when $d=2$ results in the firefighters being able to contain the fire. This is despite the fact that it appears to be impossible for two firefighters to contain the fire on the infinite square grid when $d=2$, which we formalize in Conjecture <ref>.

Two firefighters do not suffice to contain the fire on the infinite square grid when $d=2$.

We further expand upon Conjecture <ref> below in Conjecture <ref> using a notion from [6] with different notation. The notion of saving some portion $\rho \in [0,1]$ of the vertices with some predetermined strategy is defined as $\liminf_{n\to\infty} \frac{|B_n|}{|D_n|} = \rho$. Here $D_n$ is the set of vertices at distance $n$ or less from where the fire broke out and $B_n$ is the set of vertices at distance $n$ or less from where the fire broke out that will eventually burn.

Two firefighters cannot save more than $\frac34$ of the vertices on the square grid when $d=2$.

§ STRONG GRID

One of the main results from [2] shows that eight firefighters are necessary on the strong grid when $d=1$ (i.e. $f_1(G_{\boxtimes}) = 8$), and the strategy illustrated in Figure <ref> shows that $f_2(G_{\boxtimes}) \leq 4$. It is also known that in the original game four firefighters is the minimum number of firefighters where the fire can be contained on the strong grid [8]. Thus the firefighters cannot contain the fire with three firefighters at any distance. If the distance is increased, the same strategy can be used to contain the fire, so all distances greater than two also require four firefighters.

Four firefighters are necessary and sufficient to contain the fire on the strong grid when $d=2$.

Refer to Figure <ref> to see that four firefighters are sufficient[There are instances in Figure <ref> where a vertex will have a firefighter on that vertex on multiple turns. In these cases we colour the inside of the vertex the colour the second firefighter would have, and the outside of the vertex is coloured the colour of the initial firefighter. This could cause issues if both firefighters are there on even turns or they are both there on odd turns. However, for us this does not happen so we can disregard it.]. To confirm that four firefighters are necessary, see Theorem 22 from [8] which states that three firefighters do not suffice in the original game and therefore they do not suffice for the distance-restricted game as well. Thus four firefighters are both necessary and sufficient to contain the fire on the strong grid when $d=2$.

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A strategy for containment with four firefighters on the strong grid when $d=2$. One firefighter moves across the top while the other three spiral around from the other side.

In terms of average firefighting, note that a strategy is given for the strong grid with $3+\frac{1}{T}$ firefighters for any $T \in \mathbb{Z}^+$ in [9]. That is to say, there is a strategy where whenever the turn number is $0$ (mod $T$) there are four firefighters available, and the rest of the time there are three firefighters available. This strategy almost satisfies the restrictions of the distance-restricted game when $d=2$ and only needs slight modification to satisfy the restrictions.

The initial part of the strategy has the firefighters contain the fire to a quarter plane, but then has one of the firefighters jump from being on on the right side to being on the left side. We simply modify this so that instead of maintaining the right wall and moving around the fire from top to bottom on the left, we maintain the top wall and move around the fire from right to left along the bottom. The first four moves of this are drawn below in Figure <ref>.

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The start of a strategy for containment with $3 + \frac{1}{T}$ firefighters on the strong grid when $d=2$.

§ HEXAGONAL GRID

The hexagonal grid represents an interesting challenge as the minimum number of firefighters required in the original firefighting game is the subject of Messinger's Conjecture [8]. Messinger's Conjecture states that on the hexagonal grid, one firefighter is insufficient to contain the fire. It was shown in [6] that a fire can be contained on the hexagonal grid if there is always one firefighter available except on turns $t_1,t_2$ where there is one additional firefighter. The bound was further improved to only require one extra firefighter on a single turn $t_1$ in [3]. This presents a new challenge for us as unlike previous cases we do not inherit a good lower bound from the original game.

First we will observe another interesting difference about the distance-restricted game in comparison to the original game. If we consider the strategies for containing the fire with one firefighter plus an extra firefighter on an extra turn or two and try to find a distance that will always allow the strategy to be transferred, then we will quickly see that a problem arises. Namely, the distance becomes a monotonic increasing function of the turn number on which the extra firefighter is available. Thus no matter what distance we pick, there will be a turn number $t_1$ where if the extra firefighter comes on $t_1$ then the strategy given in [3] will violate our distance restriction. So we can see that in this case, there may be no strategy for which one firefighter with one extra firefighter or even two extra firefighters can always contain the fire for some fixed distance.

For the case of $d=1$ observe in Figure <ref> that every vertex has three paths that can be drawn out in a similar way as the four paths from the case of $d=1$ on the square grid. That is to say, a firefighter defending one of these paths cannot defend either of the other two paths. This can be formalized by saying that if the vertices are numbered (starting with the $0^{th}$ vertex as the vertex shared by the paths), then the $i^{th}$ vertices in any pair of these paths are at a minimum distance of $i$ from each other. Thus when $d=1$ three firefighters are required and clearly three firefighters suffice.

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The three paths used to show that two firefighters do not suffice on the hexagonal grid when $d=1$

Now observe that in the case of $d=2$ the fire is contained in five turns as illustrated in Figure <ref>. For any distance greater than $2$ the same strategy can be applied and thus two firefighters are also sufficient when $d > 2$.

Two firefighters suffice to contain the fire on the hexagonal grid when $d \geq 2$.

See Figure <ref>.

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A strategy for containment with two firefighters on the hexagonal grid when $d=2$

We also conjecture that two firefighters are necessary to contain the fire in the case of $d=2$ which we formalize in Conjecture <ref>. This conjecture is a weaker version of Messinger's Conjecture, since if one firefighter is insufficient for the original game then it is certainly the case that one firefighter is insufficient for the distance-restricted game. Thus at least two firefighters would be necessary as stated in Conjecture <ref>.

One firefighter does not suffice to contain the fire in the hexagonal grid when $d=2$.

This special case of Messinger's Conjecture has a nice property which should make a proof more attainable. Namely, whenever the firefighter moves a distance of two its start and end point have exactly one neighbour in common. This common neighbour vertex now has all but one of its neighbours protected, and so this vertex is essentially protected since the fire reaching this vertex does not change anything in terms of how much the fire will spread as it cannot spread from this vertex to any new vertices. So in essence, any strategy that the firefighter employs is equivalent to protecting the vertices of some walk in the grid. It is our belief that this property could be used to show that any strategy being successful implies that a spiraling strategy must also be successful, thus allowing a proof of this conjecture to be built from a proof that the spiraling strategy does not work.

§ DISCUSSION AND OPEN PROBLEMS

Throughout this paper we have seen many upper bounds on $f_d(G,u)$ for fixed $G$ and $d$, and progress has been made towards establishing some lower bounds to complement them. Lower bounds are of particular interest in general, and especially for this game. The proofs of these lower bounds cannot make use of widely used tools like Fogarty's Hall-type condition [5] as the firefighters' positions are no longer a sequence of arbitrary moves. The way that every move affects all moves that follow can make the problem of distance-restricted firefighting more difficult than the original game.

Alongside these questions about lower bounds for particular graphs, there are also questions about the game in general. For example we initially wondered if every infinite, planar, $k$-regular, vertex-transitive graph would require $k$ firefighters to contain the fire when $d=1$. However if we consider $P_{-\infty,\infty}\:\square\:C_{n}$ for any $n \in \{3,4,5,\ldots\}$ (where $P_{-\infty,\infty}$ is an infinite path that stretches out in two directions), observe that two firefighters can contain the fire by starting far enough away from the fire along $P_{-\infty,\infty}$ in either direction and then forming a protective barrier in the form of two $n$-cycles. Moreover, for any vertex in the graph, the subgraph induced by the set of vertices within distance $\lfloor \frac{n-2}{2} \rfloor$ is isomorphic to the subgraph induced by the same process for any vertex in the square grid.

Another question we have is under which conditions $f_d(H,u) \leq f_d(G,u)$ for $H$ a subgraph of $G$ and $u \in V(H)$. We have seen that this does not hold in general, but if we restrict ourselves to the strong, square, hexagonal and our subdivided hexagonal grid as $G$ and $H$ then the property does hold. All these examples of the inequality not holding have to do with the fact that, unlike in the original model, the set of vertices the firefighters can eventually reach is reduced if the set of burnt vertices is a vertex cut. Due to this fact, the inequality likely involves the connectedness of both $G$ and $H$, potentially in relation to the degree of their vertices. For example, all of our grids except the subdivided hexagonal grid are $k$-regular and $k$-connected, but the subdivided hexagonal grid is not regular. So it is possible that $G$ and $H$ being $k$-regular and $k$-connected (for possibly different values of $k$) could imply that $f_d(H,u) \leq f_d(G,u)$, but a different condition than regularity would be needed for a full characterization.

We of course also have the conjectures we made through the paper which we reiterate here.

We consider Conjectures <ref> and <ref> to be the more interesting conjectures as Conjecture <ref> is something that would likely be proven in order to then prove the other two conjectures from Section <ref>. Even if Conjecture <ref> is unsolved, Conjecture <ref> would still be an interesting result as it would represent a new proof of a lower bound in firefighting that is non-trivial and does not use Fogarty's Hall-type condition.

Conjecture <ref> would also be an interesting result, especially if Messinger's conjecture was shown to be false. If Messinger's conjecture were false, Conjecture <ref> would remain open and would open up a new question of which values of $d$ permit a strategy where one firefighter can contain the fire on the hexagonal grid. Alternatively, proving Conjecture <ref> would imply that any counterexample to Messinger's conjecture would require the firefighter to move a distance greater than $2$ at least once. In any case, Conjecture <ref> could lead to some interesting explorations into how the game behaves when the firefighters' strategy can be considered as a walk.

§ ACKNOWLEDGEMENTS

Authors A.C. Burgess and D.A. Pike acknowledge research support from NSERC Discovery Grants RGPIN-2019-04328 and RGPIN-2016-04456, respectively.

§ ADDITIONAL STATEMENTS

Competing interests: The authors declare none.
Data availability: There is no data associated with this article.

[1]
Xujin Chen, Xiaodong Hu, Changjun Wang, and Ying Zhang, Continuous
firefighting on infinite square grids, Theory and applications of models of
computation, Lecture Notes in Comput. Sci., vol. 10185, Springer, Cham, 2017,
pp. 158–171. 3655493

[2]
Sarah Days-Merrill, Firefighter problem played on infinite graphs,
Honors thesis, Bridgewater State University, 2019.

[3]
Abdullah Dean, Sean English, Tongyun Huang, Robert A. Krueger, Andy Lee, Mose
Mizrahi, and Casey Wheaton-Werle, Firefighting on the hexagonal grid,
Discrete Appl. Math. 305 (2021), 16–22. 4309398

[4]
Stephen Finbow and Gary MacGillivray, The firefighter problem: a survey
of results, directions and questions, Australas. J. Combin. 43
(2009), 57–77. 2489409

[5]
Patricia Fogarty, Catching the fire on grids, Masters thesis, University
of Vermont, 2003.

[6]
Tomáš Gavenčiak, Jan Kratochvíl, and Paweł Prałat, Firefighting on
square, hexagonal, and triangular grids, Discrete Mathematics 337
(2014), 142–155.

[7]
Bert Hartnell, Firefighter! An application of domination, 25th
Manitoba Conference on Combinatorial Mathematics and Computing, 1995.

[8]
Margaret-Ellen Messinger, Firefighting on infinite grids, Masters
thesis, Dalhousie University, 2005.

[9]
Margaret-Ellen Messinger, Average firefighting on infinite grids,
Australas. J. Combin. 41 (2008), 15–28. 2416961

[10]
Connor Wagner, A new survey on the firefighter problem, Masters thesis,
University of Victoria, 2021.

[11]
Ping Wang and Stephanie A. Moeller, Fire control on graphs, J. Combin.
Math. Combin. Comput. 41 (2002), 19–34. 1899921

§ W

The more interesting case with $d=3$ is if we allow the fire to burn for one turn before the two firefighters start defending vertices. If we attempt to use the same strategy of spiraling around the fire, we can equate the problem of whether or not the fire will be contained with a problem related to the growth of a recurrence that we will derive from the strategy. In order to set up this problem we first have to derive the number of turns it takes for two $d=3$ firefighters to cover[A corner is covered when the firefighters change directions] the first $k$ corners. We can derive this as a recurrence relation by considering the number of turns it takes to go from the $(k-1)^{th}$ corner to the $k^{th}$ corner, represented as $t_k-t_{k-1}$. First we will note that it takes $3 + 3t_{k-6} + 2(k-6)$ turns for the fire to reach the $(k-6)^{th}$ corner. In order to understand this we will observe Figure <ref>.

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An illustration of how the fire moves along the outside of the firefighters

In Figure <ref> we observe the fire (initially at the top left red vertex) moving along the edge created by the firefighters (the alternating black and gray vertices). We see that for each firefighter turn there is a red vertex that is in the same position relative to the firefighter's positions. These red vertices are all $3$ apart, so the fire will take $3$ turns per firefighter turn to move along this edge. However the green vertex is in the same position as the initial red vertex, but relative to the next edge the firefighters have created. Thus the fire does not need to get to the last red vertex, it needs get to the green vertex. As a result it takes the fire $2$ extra turns for every corner that it passes. From here we combine these two pieces to see that the fire will take $3$ turns per firefighter turn($3t_{k-6}$), plus $2$ turns per corner($2(k-6)$), plus however many turns it takes the fire to get to the initial red vertex relative to the first edge. We now observe Figure <ref>, representing an initial configuration for the game, to see that it takes $3$ turns for the fire to get to the desired vertex, represented by the green vertex, which gives us a $+3$ on the end of our expression. Thus we see that to reach the $(k-6)^{th}$ corner it takes $3t_{k-6} + 2(k-6) + 3$ as desired.

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An illustration of how the fire gets to the desired initial position

The fire will then have to burn for $2(t_{k-5} - t_{k-6}) + 1$ turns to have a clear path straight towards the vertex outside the $(k-5)^{th}$ corner. We can see this by noting that the firefighters move $2$ towards the $(k-5)^{th}$ corner each turn in the direction we want. We need to go one past this position, which gives us $2(t_{k-5}-t_{k-6})+1$. This is also along the line towards the $(k-1)^{th}$ corner, so the downward[By downward here we mean in reference to the rotation of the game that gives the same concept as in Figure <ref>] growth of the fire from here is exactly how far away the corner will be when the firefighters reach the $(k-1)^{th}$ corner.

Now until turn $t_{k-1}$ the fire is burning in this downward direction enlarging the distance between the corner $k-1$ and the corner $k$ until the firefighters reach corner $k-1$. So we take the two quantities, $3t_{k-6} + 2(k-6) + 3$ and $2(t_{k-5} - t_{k-6}) + 1$, and subtract them from $t_{k-1}$ to get our formula for $t_k - t_{k-1}$ and the above formula follows from there. A reference image is shown in Figure <ref> to aid in understanding all of this.

This recurrence along with the initial conditions $t_1=2,t_2=5,t_3=11,t_4=22,t_5=41,t_6=74$ counts the desired number of turns. If we can show that $t_k \geq 3 + 3t_{k-3} + 2(k-3)$ then the firefighters can never catch up to the `furthest around' firefighters and thus can never contain the fire. We also conjecture that if the recurrence ever begins to decrease or becomes negative then the fire is contained using this strategy(I should probably make an actual proper conjecture here).

(-1,-1) – (0,0);
(0,0) – (5,-5) ;
(5,-5) – (4,-6);
[red,dashed,thick] (0,0) – (5,0) node[midway,fill=white,above] $2(t_{k-5} - t_{k-6})$ turns;
[red,thick,dashed,->] (5,0) – (5,-5) node[midway,fill=white] “Downwards Growth”;
[thick,->] (-2,0) .. controls (-1,0.5) .. (0,0);
at (-2,0) [circle,fill=white,label=left:$3t_{k-6} + 2(k-6) + 3$ turns];
at (0,0) [circle,draw,minimum size=0.5mm,label=above:$t_{k-6}$] ;
at (5,-5) [circle,draw,minimum size=0.5mm,label=right:$t_{k-5}$] ;

A sketch of how the recurrence relation is derived

It is also worth noting that if we change our initial conditions to $t_1=1,t_2=2,t_3=4,t_4=7,t_5=11,t_6=16$, which matches up with the case of a single fire and two firefighters, then the sequence stops growing after turn $8$ and actually becomes negative. This is the only piece of evidence we have about how the recurrence behaves in a situation where the fire is contained.

We made attempts to solve the recurrence in order to see how it grows, but solving it by hand would be a serious undertaking of its own and when we gave it to a python solver to solve it ran for 90 days without coming to a solution. We also made attempts to just prove that the growth of the function was bounded below by a sufficiently large exponential function, which were also not successful.

What we did get from these attempts at solving the recurrence is the general form of the solution(displayed below in Equation <ref>), as well as observing numerically that the recurrence appears to tend to $\phi^k$ as $k$ gets large. We also have some numerical solutions to the general form of the recurrence, but we do not know enough about how these numerical solutions compare to the actual solution to make any inferences about how our numerical solutions compare to the original solution.

\begin{equation}\label{eqn:gensoln}
t_k = A(i)^k + A(2k+4) + B(-i)^k + \frac{C(A+Bk)}{A}\Bigg(\frac{1-\sqrt{5}}{2}\Bigg)^k + \frac{D(A+Bk)}{A}\Bigg(\frac{1+\sqrt{5}}{2}\Bigg)^k
\end{equation}


# Photonic and phononic modes in acoustoplasmonic toroidal nanopropellers

Beatriz Castillo López de Larrinzar  Instituto de Micro y Nanotecnología IMN-
CNM, CSIC, CEI UAM+CSIC, Isaac Newton 8, Tres Cantos, 28760 Madrid, Spain
Jorge M. García  Instituto de Micro y Nanotecnología IMN-CNM, CSIC, CEI
UAM+CSIC, Isaac Newton 8, Tres Cantos, 28760 Madrid, Spain  Norberto Daniel
Lanzillotti-Kimura  Université Paris-Saclay, CNRS, Centre de Nanosciences et
de Nanotechnologies, 91120 Palaiseau, France  Antonio García-Martín
Corresponding author<EMAIL_ADDRESS>Instituto de Micro y
Nanotecnología IMN-CNM, CSIC, CEI UAM+CSIC, Isaac Newton 8, Tres Cantos, 28760
Madrid, Spain

###### Abstract

Non-conventional resonances, both acoustic and photonic, are found in metallic
particles with a toroidal nanopropeller geometry that is generated by sweeping
a three-lobed 2D-shape along a spiral with twisting angle, $\alpha$. For both
optical and acoustic cases, spectral location of resonances experiences a red-
shift as a function of $\alpha$. We demonstrate that the optical case can be
understood as a natural evolution of resonances as the spiral length of the
toroidal nanopropeller increases with $\alpha$, implying a huge helicity
dependent absorption cross section. In the case of acoustic response, two red-
shifting breathing modes are identified. Additionally, even small $\alpha$
allows the appearance of new low-frequency resonances, whose spectral
dispersion depends on a competition between length of the generative spiral
and the pitch of the toroidal nanopropeller.

Keywords: Acoustoplasmonics, Nanopropeller, Chirality, Broken symmetries

##  1 Introduction

The engineering of nanoacoustic and nanophotonic structures is relevant for
the conception of novel optomechanical systems, for quantum and sensing
technologies, as well as for telecommunications, among others [1, 2, 3]. The
size, shape, and materials of a nanostructure determine simultaneously its
elastic and electromagnetic behaviors. The development of nanofabrication
techniques over the last thirty years has enabled access to nanostrutures with
unprecedented length scales, frequencies, and resolution [4, 5, 6, 7, 8].

Objects are considered chiral when their mirror images cannot be superimposed,
even after spatial rotations in three dimensions. The importance of chiraly is
such that it is critical in life systems since biologically relevant molecules
are predominantly chiral (proteins, amino acids, etc.) [9]. Handedness (right
vs. left, dextro vs. levo) identifies the two different enantiomers in a
chiral structure, and it is the crucial element determining how systems
interact with their environment [10].

Thus, the presence of chirality in a system, unequivocally leads to a
geometrical broken symmetry. Good examples of macroscopic systems are helices,
nuts or screws. Particularly interesting is the nut-screw pair. In resemblance
to what happens with a screw with the right helicity (and pitch) that nicely
couples with the complementary nut, and does not with the other, circularly
polarized light interacts with a chiral structure in a different manner
depending on the helicity (left or right, levo or dextro) of its circular
polarization. In resonant systems, such as metallic ones that support plasmon
excitations, the optical interaction with the nanostructure is drastically
enhanced. Therefore, the differences in the optical response for levo or
dextro light incidence (known as Chiro-Optical Effect - COE) become prominent
[11, 12, 13] . Among the plethora of structures to be employed for chiral
applications, toroidal propeller-like structures present peculiar properties
such as COE reversal [14].

Macroscopic toroidal propellers are used in aviation and maritime transport to
implement less noisy propulsion systems, as they lead to a reduction of tip
cavitation and fluid vortices creation since the propeller have no tips.
Aquatic toroidal propellers are also improving the fuel efficiency. Although
in the aforementioned cases the actual fabrication of these structures by
means of classical machining methods is complex, new additive manufacturing
techniques may soon charge drastically this scenario. In the nanoscopic world,
advanced nanofabrication tools have enabled the possibility to fabricate these
systems by means of scanning electron lithography with an unprecedented
resolution and accuracy [15, 16, 14].

The development of novel nanostructures possessing simultaneous optical and
acoustic resonances is at the core of active fields of research such as
optomechanics and nanophononics. In this sense, most of the reported systems
rely on relatively simple systems (multilayers, plasmonic antennas, dielectic
resonators), [17, 18, 19, 20, 21, 22], where chirality remains elusive.

In this manuscript we develop a complete theoretical study of the optical and
acoustic characteristics of a three-lobed toroidal nanopropeller (TnP) made of
gold. We do so via the analysis of the scattering and absorption cross
sections for the optical response, and calculating the average displacement of
the material for an harmonic response after a 5 degree Kelvin thermal uniform
expansion for the acoustic one. We demonstrate that the spectral location of
the optical plasmon resonances red-shift as a function of the twist angle
$\alpha$ of the propeller. Additionally we observe an interesting interference
effect for $\alpha$ in the range of 90 deg. to 270 deg., where the
electromagnetic field profile is highly dependent on the spin or helicity of
the incident wave. When analyzing the acoustic response we find an even richer
and more intriguing behavior. Through the analysis of the spectra of the
average RMS displacement we demonstrate that the resonance modes supported by
the untwisted system also experience a red shift as the twist angle increases.
But, at variance with the optical case, the system also allows unexpected new
resonant modes at lower energies only present for non-zero twisting angles.

##  2 Materials and Methods

The geometry of the TnP used in this study is based on a three-lobed structure
of identical rings, with an inner diameter of 65 nm and a wall thickness of 20
nm, as presented in Figure 1. The centers of the rings are disposed in the
vertices of an equilateral triangle, as a result, the whole structure is
circumscribed in a circle with a diameter of 200 nm. The twisted TnP is
subsequently obtained by vertically sweeping this geometry along a helical
path (L) with a varying $\alpha$. That sweep, together with the vertical
dimension (H=60 nm), defines the pitch of the propeller ($P=2\pi H/\alpha$).
TnP is a singularly connected structure and with a topological genus 3.

Figure 1: Schematic view of a $\alpha=60$ degrees twisted gold TnP. (a) Top-
view depicting the geometrical parameters. The nanostructure is inscribed in a
circle of D=200 nm diameter, the diameter of the inner hole is 65 nm and the
wall thickness is 20 nm. Label A corresponds to breathing modes resonances
(see section 3.2) (b) The height is H=60 nm, $\alpha$ is the twisting angle.
The length of the helical path (blue line) $L$ increases from H (for
$\alpha=0$) to $L$ as the twisting angle $\alpha$ increases (see Eq. 2).

As mentioned, to simulate the optical and acoustic response, we assume the TnP
is made of gold, following the dispersive optical constants given in Ref. [23]
provided in the Lumerical® software, and the thermal expansion, sound speed
and Young modulus embedded in the Finite Element Method COMSOL®.

Lumerical® software suite has been employed to solve the scattering and
absorption cross section. The solutions are based on the Finite Difference
Time Domain (FDTD) method. We use an impinging circularly polarized plane wave
with the proper circular polarization along the z axis, with a normalized
amplitude in the whole simulation cell. The cell geometry (4 $\mu$m × 4 $\mu$m
× 7 $\mu$m) ensures that perfectly absorbing boundary conditions do have a
negligible effect on the electromagnetic fields obtained. We use a refined
mesh in the nanostructures and in the near field region (0.3 $\mu$m × 0.3
$\mu$m × 0.9 $\mu$m) of 2 nm × 2 nm × 2 nm, dx-dy-dz, respectively, growing
uniformly up to a maximum of 40 nm out of the near field close to the
simulation boundaries, so that convergence (to the best of our numerical
capabilities) is attained. The total and scattered fields are then collected
to give rise to intensity color maps and cross sections.

The Finite Element Method (FEM) within COMSOL® provides a suitable environment
to model the acoustic response upon isotropic heat expansion of a metallic
nanostructure. In our simulations the toroidal propeller is, as mentioned
above, assumed to be made of gold, standing on a layer of silicon dioxide
semi-spherical substrate. The material parameters used for FEM simulation are
the ones built-in the materials library of COMSOL®. The SiO2 substrate is a
800 nm diameter semi-spherical region surrounded by perfectly matching layers
(PML) that truncate the physical domain, see e.g. Ref. [20].

##  3 Results

###  3.1 Optical characteristics

As it can be seen, in a twisted TnP geometry like the one depicted in Figure
1b, there is a clear handedness. Therefore, it is expected that the optical
response will present an helicity dependence that should, additionally, depend
on the twisting angle $\alpha$. To explore that possibility, the optical
response of this structure is interrogated by impinging with a plane wave
whose propagation vector is perpendicular in the x-y plane (i.e. in the z
direction, from $z=+\infty$), and its polarization is either left-circular
(LCP) or right-circular (RCP) as

$|LCP\rangle=\frac{|x\rangle+i|y\rangle}{\sqrt{2}};\quad|RCP\rangle=\frac{|x\rangle-i|y\rangle}{\sqrt{2}},$
(1)

where $|x\rangle$ represents a linearly polarized wave along the x-direction
while travelling along the z-direction, and $|y\rangle$ is travelling in the
same direction but linearly polarized along the y-direction.

Let us first describe the optical response, in the form of scattering and
absorption cross sections, for the untwisted nanostructure as a function of
the wavelength in the visible range of the optical spectrum. Similarly to a
system resembling a metallic gold disc [24], the system present a plasmon
resonance in the range of 0.8 $\mu$m (see Figure 2a). Another smaller
resonance is also observed around 0.6 $\mu$m, expected as long as the
dielectric constant of Au stays in the metallic range. It is worth noting that
for this case there is no difference between LCP and RCP, as foreseen for a
structure with no handedness.

Figure 2: (a) Scattering and absorption cross section for the untwisted TnP in
Figure 1 for LCP and RCP (no differences for this geometry), depicting clearly
the main plasmonic resonance at $\lambda_{\text{res}}=0.825~{}\mu$m, and other
weaker at shorter $\sim 0.6\mu$m wavelengths. (b) Scattering and (c)
absorption cross sections for a fixed wavelength, $\lambda_{\text{res}}$,
varying the the twisting angle.

In Figure 2(b) and (c) we present the scattering and absorption cross sections
for the resonance at wavelength $\lambda_{\text{res}}=0.825~{}\mu$m, as a
function of the twisting angle. Noticeably, in this case there are big
differences between LCP and RCP incoming waves, signaling the presence of
chirality. However, the most prominent feature is the oscillatory pattern in
optical response. In this structure, the only relevant parameter, from the
point of view of the optical resonances, that changes with the twisting angle
is the length of the pores forming the lateral surface of the propeller. The
length of the helix used for generating the TnP (see Figure 1(b)) can be
expressed as

$L=\sqrt{\left(\frac{\alpha D}{2}\right)^{2}+H^{2}},$ (2)

where $\alpha$ is in radiands, D is the radius of the helix, and H the height
as mentioned above (see Figure 1b). Notice that the radius of the helix,
optically speaking, will have a different dimension depending on the nature of
the resonance, i.e. weather the field is predominantly in the air or in the
metal.

In order to complete the optical analysis, it is apparent that the complete
spectrum of the scattering and absorption cross sections needs to be obtained.
This is shown in Figure 3, where we present the scattering and absorption
cross sections as a function of the wavelength, for twisting angles bellow 360
degrees.

In all panels in Figure 3 we observe that the resonances at 0.825 $\mu$m and
at 0.6$\mu$m, as a function of the twisting angle, behave in the following
manner: for $\alpha$ from 0 to c.a. 50 deg. there is very little dispersion,
for values from 50 to $\sim$110 deg. the resonances strongly shifts to longer
wavelengths, and, interestingly, for twisting angles $\alpha\gtrsim$120 deg.
they disperse linearly towards high wavelengths running parallel to each
other. This happens both for the absorption and the scattering cross sections.
Noticeably, for the twisted structure ($\alpha\sim 120$ degrees), the
absorption (depicted in (c) and (d)) is over two times bigger than for the
untwisted one, but the volume of the system remains essentially the same.
There are obvious differences in the intensity for LCP and RCP waves, pointing
out COE, but, apart from the actual value of the cross sections, the overall
oscillatory phenomenology is virtually equivalent for both helicities. In
fact, the plot of Equation 2 (see Appendix Appendix A: Length of the helix)
shows that it accurately describes the trend followed by the resonance peaks,
even in the fact that some may have different slopes (that would depend on a
varying effective D).

Figure 3: Top: Scattering cross sections for LCP (a) and RCP (b) as a function
of the twist angle and of the wavelength. Bottom: Absorption cross sections
for LCP (c) and RCP (d) as a function of the twist angle and of the
wavelength. As it can be seen, there is an almost linear dependence of the
resonances for increasing twist angles. The high helicity dependence, COE, of
the absorption is clearly observed in the bottom panels.

Additionally, from a simple inspection of the panels in Figure 3 there are
obvious signatures of a sizeable COE. This COE is specially relevant in the
absorption cross section, where the response of the propeller is enhanced for
LCP waves respect to RCP ones, reaching differences larger than 100%. For the
scattering cross section, the differences are not as marked and occur at
shorter wavelengths than for the absorption.

###  3.2 Acoustic characteristics

Let us now address the characterization of the acoustic response of the TnP.
For that end we study the vibrations, characterized by the displacement of
each point in the harmonic regime upon an initial isotropic thermal expansion
produced by a $\Delta T=5$K.

Figure 4: Average RMS displacement of a gold TnP as a function of the acoustic
frequency for different twisting angles (0 to 60 degrees). The curves for non-
zero twist angle are shifted 0.015 nm vertically for clarity. The peaks
represent acoustic phononic modes of the structure. In the inset we present
the angular evolution (1 to 15 degrees) of the low frequency modes that appear
only for non-zero twist angles.

In order to find the resonant modes corresponding to the structure, we monitor
the average displacement (i.e. the 3D integral of the RMS displacement of each
point of the TnP normalized to its volume). The resonant modes appear as
distinct peaks as presented in Figure 4. Two natural, low frequency, modes for
an untwisted (red curve) TnP appear at 4.6 GHz and 5.5 GHz respectively and
correspond to the breathing modes with maximum (low frequency) and minimum
(high frequency) displacement at the point marked as A in Figure 1. As soon as
the structure is slightly twisted (even just one degree) we clearly observe
two different phenomena. The first one is that the resonances associated with
the breathing modes shift towards lower energies (i.e. the wavelength
associated to the resonances increases), in perfect correspondence with the
previously presented optical behavior. The second observation, now at variance
with the optical case, is that in the 3 to 4 GHz region two new resonances
appear that disperse downshift in frequency as the twist angle increases. For
zero twist angle (red curve in Figure 4 and its inset) there are no resonances
in this region, but even for a twist of one degree, they appear (black curve
inset Figure 4). The evolution with twist angle from $\alpha$=0 to $\alpha$=15
deg. (see inset in Figure 4) shows a further increase in intensity and a
downshifting in frequencies. In Figure 7 in Appendix Appendix B: Displacement
profiles we present some snap-shots for the breathing modes at $\alpha=0$ deg.
and for the low frequency twist-enabled modes at $\alpha=15$ deg..

Figure 5: Colormap of the average RMS displacement of a gold TnP as a function
of the acoustic frequency for one complete loop of the twisting angle. The
intensity is saturated at 0.05 nm for clarity, and the peaks representing the
five lowest acoustic phononic modes of the structure are marked with dots. The
insets show the geometry for a TnP with a twist angle of 360 degrees (a), 120
degrees (b) and untwisted (c).

Similarly to the optical case, we are presenting in Figure 5 the whole
evolution of the resonance location for a TnP with a complete twist angle (360
deg.). As already indicated in Figure 4, we can see that the lowest energy
resonances for the untwisted geometry (inset (c) of Figure 4) appear for 4.6
GHz and 5.5 GHz and evolve towards lower energies as the twist angle
increases. One relevant aspect that can be extracted from this figure is that,
although the resonances always evolve towards a lower energy value, there is a
change in the trend. Initially, the frequency decreases roughly as 1/L as
expected to fit with the increase of helix lenght (see Equation 2). However,
this trend changes at $\alpha\sim 120$ deg.. To shed some light into the
change in slope, we refer to the features depicted in the insets (a) and (b)
in Figure 5, showing the geometry for twist angles of 360 degrees and 120
degrees respectively. If the structure were made by a single, isolated lobe,
at one full loop ($\alpha$=360 deg.) the height of the TnP would match the
pitch of the helix, and that would be the first occasion in which one lobe in
the base would have another one exactly above. However, our structure is made
of three lobes, and Figure 5a, resembles a much more twisted structure, not
surprisingly it appears to have three twists. In fact, as seen in Figure 5b,
it is at one third of a twist ($\alpha=120$ deg.) when a lobe of the base have
another one above. It is then when two effects compete: the resonance trend to
decrease in energy due to the growth of L against the resistance due to the
upper and lower branches of the TnP.

Eventually, for a very twisted geometry, the resonance should converge to that
of a disk shape with a twisted void geometry inside, so compressed that the
variation of the angle should have a negligible effect.

##  4 Discussion and conclusions

Understanding the origin of these optical and acoustic modes is essential to
actually use them in optomechanical systems, where the helicity of light can
be use as a design parameter. Towards that end, in this work we have presented
a complete study of the optical and acoustic properties of a three-lobed
twisted toroidal nanoscale propeller-like structure. The presence of the twist
in the TnP is indicated by a red-shift in both its optical and acoustic
resonances. This red-shift occurs because the wavelength couples with the
effective length of the generative helix, which varies with the twist angle of
the TnP. Consequently, the twist angle acts as a tuning structural parameter
that influences the nature, intensity (which depends on helicity), and
spectral positions of the optical and acoustic resonances. Additionally, the
twist present in the TnP leads to new low frequency resonant acoustic modes
that are not present in the spectrum of an equivalent untwisted structure.
These new acoustic modes appearing at non-zero twisting angles enable a new
knob in the engineering of opto-phononic interactions at the nanoscale.

## Appendix A: Length of the helix

Figure 6: Relationship of the variation of the helix length ($\Delta$L) of the
TnP helix as a function of the twisting angle for different values of the
parameter D.

In the main text we have argued that the trend followed by the optical and
acoustic resonances might be caused by the change in the length of the helical
path as the angle increases. This is based on the well-known fact that
resonances occur whenever the wavelength is commensurate with a representative
length of the system. In our case, the aspect ratio of the TnP changes with
$\alpha$ by varying L according to Equation. 2. In Figure 6, we present the
behavior of the red-shift that the resonances should experience if the only
aspect to be taken into account is the variation of L. As it can be seen, the
graph qualitatively describes the physical picture presented in Figures 3 and
5. Obviously, this has to be taken just as a poor-man phenomenological hand-
waving explanation of the actual physics, since there are other aspects
playing a key role. Some of them are the dispersive nature of the dielectric
permittivity of the materials or the actual path followed by the mode (i.e.
where the field or dispersion is maximal), the latter would modify the
effective length of the helix, giving rise to different slopes, as seen
varying D.

## Appendix B: Displacement profiles

Figure 7: (a) and (b) RMS displacement spatial profile of the two main
acoustic modes for the TnP without torsion. (c) and (d) RMS displacement for
the twist-allowed modes for a TnP with $\alpha=$15 degrees. The displacement
is represented by the color scale and the deformation in the has been scaled
for a better view.

In Figure 7a and b, we present the displacement spatial profiles corresponding
to the two lowest energy modes for the untwisted TnP, appearing at 4.6 and
5.48 GHz, respectively (see red lines in Figure 4). We can clearly see that
the profiles correspond to two complementary breathing modes: the one with
lowest energy corresponds to maximum displacement at the ”tip” of each lobe,
marked as A in Figure 1, whereas the one at highest energy essentially
presents a anchor point at that location.

In Figure 7c and d, we depict the displacement profiles for a TnP with
$\alpha=15$ deg.. In this case the deformation is essentially devoted to exert
a torsion in the TnP. The anchor point at Figure 7c is located at the
intersections of the lobes, and at the center of the TnP, allowing more
freedom of movement than in the case presented in Figure 7d. In that case,
there is anchor point locate close to the edge of the lobes, resulting then in
a bigger frequency.

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