Related papers: Cops and robbers on $2K_2$-free graphs

We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq \alpha(G) \leq…

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the…

We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove…

We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by…

In the game of \emph{cops and robbers} on a graph $G = (V,E)$, $k$ cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber's turn, he moves similarly. The cops win if there is…

We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any $n$-vertex…

For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that…

The game of Cops and Robbers on graphs is a well-studied pursuit--evasion model whose central parameter, the cop number, captures the minimum number of pursuers required to guarantee capture of an adversary on a given graph. While the cop…

We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by $O(\sqrt{n})$,…

(abstract shortened to meet arxiv's length requirements) We investigate two variants of the classical Cops and robber game in graphs, recently introduced by Lee, Mart\'inez-Pedroza, and Rodr\'iguez-Quinche. The two versions are played in…

Various models to quantify the reliability of a network have been studied where certain components of the graph may fail at random and the probability that the remaining graph is connected is the proxy for reliability. In this work we…

Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin…

In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G. The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is…

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and…

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence…

In this paper, we determine the graphs with maximum value of the sum number from $k$-clique spectral radius to $(2r-1)$-clique spectral radius among all $2K_{r}$-free graphs on $n$ vertices for $ r\le k$ and large $n$. We also determine the…

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop…

The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a…

Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber. In alternating turns, each player moves (all) their figures. The cops try to capture the…

Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token…