Related papers: Cops and Robbers on Graphs with Path Constraints

The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set…

It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with…

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…

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved…

Cops and Robbers is a well-studied pursuit-evasion game in which a set of cops seeks to catch a robber in a graph G, where cops and robber move along edges of G. The cop number of G is the minimum number of cops that is sufficient to catch…

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…

In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. The throttling number $th_c(G)$ of a graph $G$ is the minimum possible value of $k…

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…

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 prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game which is more symmetric called the cop and killer game. We prove for all $c < 1$ that…

Cops and robbers is a turn-based pursuit game played on a graph $G$. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number $c(G)$ denotes the minimum number of…

Cops and Robbers is a type of pursuit-evasion game played on a graph where a set of cops try to capture a single robber. The cops first choose their initial vertex positions, and later the robber chooses a vertex. The cops and robbers make…

In their 1997 paper titled ``Fruit Salad", Gy\'{a}rf\'{a}s posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that…

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber…

A 2018 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \bmod k$ as the complete graph on $k+1$ vertices.…

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical…

We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and…

Given a graph $G$, let $\Delta_2(G)$ denote the maximum number of neighbors any two distinct vertices of $G$ have in common. Vu (2002) proposed that, provided $\Delta_2(G)$ is not too small as a proportion of the maximum degree $\Delta(G)$…

A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic…

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb{N}$, if $G$ is a…