Throttling for the game of Cops and Robbers on graphs

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k+{\mathrm{capt}}_k\left(G\right))$, where $k$ is the number of cops and ${\mathrm{capt}}_k\left(G\right)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $O\left(\sqrt{n}\right)$.     

Meyniel's conjecture holds for random graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph , which improves upon existing results showing that asymptotically almost surely the cop number of is provided that for some . We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. This will also be used in a separate paper on random d‐regular graphs, where we show that the conjecture holds asymptotically almost surely when . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396–421, 2016     

Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Large classes of infinite k‐cop‐win graphs

While finite cop‐win finite graphs possess a good structural characterization, none is known for infinite cop‐win graphs. As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number. More precisely, for each infinite cardinal κ and each positive integer k, we construct 2^κ non‐isomorphic k‐cop‐win graphs satisfying additional properties such as vertex‐transitivity, or having universal endomorphism monoid and automorphism group. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 334–342, 2010     

Bounds on the length of a game of Cops and Robbers

In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph $G$. All players occupy vertices of $G$. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on $G$ is the cop number of $G$, denoted $c\left(G\right)$, and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an $n$-vertex graph with cop number $k$ is $O\left(n^{k+1}\right)$. More recently, Bonato et al. (2009) and Gaven?iak (2010) showed that for $k=1$, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within $n?4$ rounds. In this paper, we show that the upper bound is tight when $k\ge 2$: for fixed $k\ge 2$, we construct arbitrarily large graphs $G$ having capture time at least ${\left(\frac{\left|V\left(G\right)\right|}{40k^4}\right)}^{k+1}$.In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether $k$ cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether $k$ cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of Goldstein and Reingold (1995), which was recently proved through other means (Kinnersley, 2015). We also show that $n$-vertex strongly-connected directed graphs with cop number 1 can have capture time $\Omega \left(n^2\right)$, thereby showing that the result of Bonato et al. (2009) does not extend to the directed setting.     

The impact of loops on the game of cops and robbers on graphs

In the game of cops and robbers on graphs, the cops and the robber are allowed to pass their turn if they are located on a looped vertex. This paper explores the effect of loops on the cop number and the capture time. We provide examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play.     

A note on degree-continuous graphs

The minimum orders of degree-continuous graphs with prescribed degree sets were investigated by Gimbel and Zhang, Czechoslovak Math. J. 51 (126) (2001), 163–171. The minimum orders were not completely determined in some cases. In this note, the exact values of the minimum orders for these cases are obtained by giving improved upper bounds.     

A note on compact graphs

An undirected simple graph G is called compact iff its adjacency matrix A is such that the polytope S(A) of doubly stochastic matrices X which commute with A has integral-valued extremal points only. We show that the isomorphism problem for compact graphs is polynomial. Furthermore, we prove that if a graph G is compact, then a certain naive polynomial heuristic applied to G and any partner G′ decides correctly whether G and G′ are isomorphic or not. In the last section we discuss some compactness preserving operations on graphs.     

A note on assymmetric graphs

We describe a partition of the points of a graph which is related to its automorphism group. We then prove that the group of a tree is trivial if and only if this partition is the trivial one, and we formulate an algorithm which produces such a partition. Some application to graphs in general are also considered. Work supported in part by NSF Grant GP 11618.     

A note on n-extendable graphs

A graph G having a perfect matching is called n-extendable if every matching of size n of G can be extended to a perfect matching. In this note, we show that if G is an n-extendable nonbipartite graph, then G + e is (n - 1)-extendable for any edge e ? E(G). © 1992 John Wiley & Sons, Inc.     

A note on perfect graphs

Let the lines of a complete graph be 3-colored so that no triangle gets 3 different colors. If two of these colors form perfect graphs then so does the third.     

A note on Ki-perfect graphs

Ki-perfect graphs are a special instance of F - G perfect graphs, where F and G are fixed graphs with F a partial subgraph of G. Given S, a collection of G-subgraphs of graph K, an F - G cover of S is a set of T of F-subgraphs of K such that each subgraph in S contains as a subgraph a member of T. An F - G packing of S is a subcollection S′? S such that no two subgraphs in S′ have an F-subgraph in common. K is F - G perfect if for all such S, the minimum cardinality of an F - G cover of S equals the maximum cardinality of an F - G packing of S. Thus K_i-perfect graphs are precisely K_i-1 - K_i perfect graphs. We develop a hypergraph characterization of F - G perfect graphs that leads to an alternate proof of previous results on K_i-perfect graphs as well as to a characterization of F - G perfect graphs for other instances of F and G.     

A note on Cayley graphs

An example is given of a finite group A of order 144, with a generating set X = {x, y} such that x³ = y² = 1 and such that the Cayley graph C(A, X) has genus 4 and characteristic −6 (both of which are small relative to the order of A), although there is no short relator of the form (xy)^r with r < 12 or of the form [x, y]^r with r < 6. Accordingly this and other possible examples do not fit into a pattern suggested by [5.], 244–268).     

A note on cospectral graphs

The notion of a (1, x) adjacency matrix is introduced, together with methods for dealing with it. It is shown that in many instances this adjacency matrix is superior to the usual (0, 1) adjacency matrix, and will distinguish cospectral pairs, when the latter will not. In those cases in which the (1, x) adjacency matrix fares no better than the (0, 1) adjacency matrix, a good deal can be said about the matrices by which the cospectral pairs are similar.