Each of several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ leads to an injective oriented colouring problem. For each case in which such a problem is solvable in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such that an oriented graph $G$ has an injective oriented colouring with the given number of colours if and only if there is no $F \in \mathcal{F}$ for which there is a locally-injective homomorphism of $F$ to $G$.

We introduce a variation of the Cops and Robber game in which the robber side consists of a robber and a decoy which are indistinguishable to the cops except under certain conditions. The cops win when one of them moves onto the same vertex as the actual robber (i.e. not the decoy) after a finite number of turns. The robber can throw the decoy to a neighbouring vertex on any turn beyond his first; such a turn for the robber consists of throwing (or dropping) the decoy and then moving. The current decoy disappears as the next is thrown so there is only a single decoy in play at any time. We characterize decoycopwin graphs in the case where the cop can distinguish between the robber and decoy only when he is on the same vertex as one of them. We also characterize such graphs if the cop can distinguish between the robber and decoy only when he has cornered at least one of them.

We introduce the bodyguard problem for graphs. This is a variation of Surrounding Cops and Robber but, in this model, a smallest possible group of bodyguards must surround the president and then maintain this protection indefinitely. We investigate some general bounds, then solve this problem for complete graphs, wheels, trees, cycles, complete multipartite graphs, and two-dimensional grids. We also examine the problem in more general Cartesian, strong, and lexicographic products. (c) 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Eternal domination is a process in which a set of guards occupying a dominating set on a graph protects against an infinite sequence of attacks. After a vertex is attacked, one guard must move along an edge to the attacked vertex and each of the remaining guards may move along an edge so that the guards again occupy a dominating set on the graph and can defend the next attack. The minimum number of guards needed in a graph Γ is the eternal domination number, denoted by γ_all^∞(Γ). In this paper, we show that the eternal domination number of a vertex-transitive graph with an efficient dominating set is equal to its domination number. We show that a Cayley graph on a generalized dihedral group whose connection set contains few or many reflections is efficiently dominated. Then, we provide an infinite family of connected Cayley graphs for which γ_all^∞(Γ) = γ(Γ)+1, generalizing a result of [Braga et al., J. Combin. Math. Combin. Comput. 96 (2016), 13–22]. Finally, we build an infinite family of connected Cayley graphs with γ_all^∞(Γ) ≥ γ(Γ)+2.

We investigate a cheating robot version of Cops and Robber, first introduced by Huggan and Nowakowski, where both the cops and the robber move simultaneously, but the robber is allowed to react to the cops’ moves. For conciseness, we refer to this game as Cops and Cheating Robot. The cheating robot number for a graph is the fewest cops needed to win on the graph. We introduce a new parameter for this variation, called the push number, which is the minimum number of cops that move onto the robber’s vertex in a game of Cops and Cheating Robot given that there are a cheating robot number of cops on the graph. After producing some elementary results on the push number, we use it to give a relationship between Cops and Cheating Robot and Surrounding Cops and Robbers. We investigate the cheating robot number for planar graphs and give a tight bound for bipartite planar graphs. We show that for a fixed k∈Z+, determining whether a graph has a cheating robot number at most k can be done in polynomial time. We also obtain bounds on the cheating robot number for strong and lexicographic products of graphs.

We look at an active version of Surrounding Cops and Robber in which the robber must never remain on their current vertex. Our parameter of interest is the active surrounding number of a graph: the minimum number of cops that suffice to surround the robber on that graph. We find exact values of our parameter for some graph families, establish bounds for others, and also explore some particular graph families for which the active surrounding number differs from the analogous parameters for several related variants.

We consider a variation of the Cops and Robbers game in which the cops do not have perfect information; the information they receive regarding the robber’s position is delayed by one round. Our parameter of interest is the time-delayed cop number of a graph G, the minimum number of cops that suffice to guarantee a win on G. We present a variety of results on this parameter, including general bounds, and make comparisons to the cop numbers of known related variants of the original game. We have particular interest in graph products, Meyniel-type bounds, and cop density.

The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different vertices cannot communicate to co-ordinate their strategy. In this paper, we extend the deduction game to allow each searcher to make k moves, where k is a fixed positive integer. We consider the value of the k-move deduction number on several classes of graphs including paths, cycles, complete graphs, complete bipartite graphs, and Cartesian and strong products of paths.

We introduce the discrete-time treatment number of a graph, in which each vertex is in exactly one of three states at any given time-step: compromised, vulnerable, or treated. Our treatment number is distinct from other graph searching parameters that use only two states, such as the firefighter problem or Bernshteyn and Lee's inspection number. Vertices represent individuals and edges exist between individuals with close connections. Each vertex starts out as compromised; it can become compromised again even after treatment. Our objective is to treat the entire population so that at the last time-step, no members are vulnerable or compromised, while minimizing the maximum number of treatments that occur at each time-step. This minimum is the treatment number, and it depends on the choice of a pre-determined length of time r that a vertex can remain in a treated state and length of time s that a vertex can remain in a vulnerable state without being treated again. We denote the pathwidth of graph H by pw(H) and prove that the treatment number of H is bounded above by ⌈1+pw(H)/r+s⌉. This equals the best possible lower bound for a cautious treatment plan, defined as one in which each vertex, after being treated for the first time, is treated again within every consecutive r+s time-steps until its last treatment. However, many graphs admit a plan that is not cautious. When r=s=1, we find a useful tool for proving lower bounds, show that the treatment number of an n× n grid equals ⌈1+n/2⌉, characterize graphs that require only one treatment per time-step, and prove that subdividing one edge can reduce the treatment number. It is known that there are trees with arbitrarily large pathwidth; surprisingly, we prove that for any tree T, there is a subdivision of T that requires at most two treatments per time-step.

Several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when $G$ is given and $H$ is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.

We explore a variant of the game of Cops and Robber introduced by Bonato et al.~where the robber is invisible unless outside the common neighbourhood of the cops. The hyperopic cop number is analogous to the cop number and we investigate bounds on this quantity. We define a small common neighbourhood set and relate the minimum cardinality of this graph parameter to the hyperopic cop number. We consider diameter 2 graphs, particularly the join of two graphs, as well as Cartesian products.

The dominating graph of a graph H has as its vertices all dominating sets of H, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of H. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to a proof that the dominating graph of a cycle on n vertices has a Hamilton path if and only if n is not a multiple of 4.

The dominating graph of a graph G has as its vertices all dominating sets of G, with an edge between two dominating sets if one can be obtained from the other by adding or deleting a single vertex of G. This is an example of a reconfiguration graph. This paper gives a brief introduction to the study of reconfiguration of dominating sets, and to the dominating graph. We highlight some previous results and present some new work. In particular we give new results on the existence of Hamilton paths in the dominating graph.

A variation of the Cops and Robber game is introduced in which the robber side consists of two robbers. The cops win by moving onto the same vertex as one of the robbers after a finite number of moves. As in the original game, the robber side can win by avoiding capture indefinitely. In this version, however, the robbers can also win by both moving onto the same vertex as the cop at the same time. Otherwise, the robbers must be located on distinct vertices. We present structural properties of ambush-copwin graphs, those graphs on which a single cop can guarantee a win. As well, we characterize ambush-copwin graphs of girth $$g \ge 4$$ and classes of ambush-copwin graphs of girth 3, particularly a class of chordal graphs.

We consider a variation of the Cops and Robber game where the cops can only see the robber when the distance between them is at most a fixed parameter ℓ. We consider the basic consequences of this definition for some simple graph families, and show that this model is not monotonic, unlike common models where the robber is invisible. We see that cops’ strategy consists of a phase in which they need to “see” the robber (move within distance ℓ of the robber), followed by a phase in which they capture the robber. In some graphs the first phase is the most resource intensive phase (in terms of number of cops needed), while in other graphs, it is the second phase. Finally, we characterize those trees for which k cops are sufficient to guarantee capture of the robber for all ℓ≥1.

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 occupying each of the robber’s neighbouring vertices. We denote by σ(G) the surrounding cop number of G, namely the least number of cops required to surround a robber in the graph G. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest include product graphs, graphs arising from combinatorial designs, and generalised Petersen graphs.

In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of arbitrary graphs G and H based on their respective clique numbers.

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in [0,1/2].

We introduce the game of Surrounding Cops and Robber on a graph, as a variant of the traditional game of Cops and Robber. In contrast to the traditional game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber’s neighbouring vertices. We denote by σ(G) the surrounding copnumber of G, namely the least number of cops required to surround a robber in the graph G. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest include product graphs, graphs arising from combinatorial designs, and generalised Petersen

For a fixed integer t, a set of vertices B of a graph G is a t-limited packing of G provided that the closed neighbourhood of any vertex in G contains at most t elements of B. The size of a largest possible t-limited packing in G is denoted Lt(G) and is the t-limited packing number of G. In this paper, we investigate the 2-limited packing number of Cartesian products of paths. We show that for fixed k the difference L2(Pk□Pn)−L2(Pk□Pn−1) is eventually periodic as a function of n, and thereby give closed formulas for L2(Pk□Pn), k=1,2,…,5. The techniques we use are suitable for establishing other types of packing and domination numbers for Cartesian products of paths and, more generally, for graphs of the form H□Pn.