For a graph $G$, the vertices of the $k$-dominating graph, denoted $\mathcal{D}_k(G)$, correspond to the dominating sets of $G$ with cardinality at most $k$. Two vertices of $\mathcal{D}_k(G)$ are adjacent if and only if the corresponding dominating sets in $G$ can be obtained from one other by adding or removing a single vertex of $G$. Since $\mathcal{D}_k(G)$ is not necessarily connected when $k < |V(G)|$, much research has focused on conditions under which $\mathcal{D}_k(G)$ is connected and recent work has explored the existence of Hamilton paths in the $k$-dominating graph. We consider the complementary problem of determining the conditions under which the $k$-dominating graph is Eulerian. In the case where $k = |V(G)|$, we characterize those graphs $G$ for which $\mathcal{D}_k(G)$ is Eulerian. In the case where $k$ is restricted, we determine for a number of graph classes, the conditions under which the $k$-dominating graph is Eulerian.

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph G, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by cc(G). We give a sufficient condition for a triangle-free graph to have attacking cop number at most 2 and we characterise when outerplanar graphs have attacking cop number 2. We also prove that all bipartite planar graphs G have cc(G) <= 4 and show this is tight by constructing a bipartite planar graph G with cc(G) = 4. Finally we construct 17 non-isomorphic graphs H of order 58 with cc(H) = 6 and c(H) = 3. This provides the first example of a graph H with cc(H) - c(H) >= 3, extending work by Bonato et al. (2014). We conclude with a list of conjectures and open problems. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph G, we are interested in b_2(G), the minimum number of rounds until the information has spread to all vertices of graph G. We are also interested in finding t_2(G), the minimum number of sources necessary so that the information spreads to all vertices of G in b_2(G) rounds. In addition to general results, we find b_2(G) and t_2(G) for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.

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.

We consider a variation of Cops and Robber, introduced in [D. Cox and A. Sanaei, The damage number of a graph, [Aust. J. of Comb. 75(1) (2019) 1-16] where vertices visited by a robber are considered damaged and a single cop aims to minimize the number of distinct vertices damaged by a robber. Motivated by the interesting relationships that often emerge between input graphs and their Cartesian product, we study the damage number of the Cartesian product of graphs. We provide a general upper bound and consider the damage number of the product of two trees or cycles. We also consider graphs with small damage number.

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance- k domination, guards initially occupy the vertices of a distance- k dominating set. After a vertex is attacked, guards “defend” by each moving up to distance k to form a distance- k dominating set, such that some guard occupies the attacked vertex. The eternal distance- k domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the situation where k=1 . We introduce eternal distance- k domination for k > 1 . Determining whether a given set is an eternal distance- k domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we use decomposition arguments to bound the eternal distance- k domination numbers, and solve the problem entirely in the case of perfect m -ary trees.

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.

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal kdomination, guards initially occupy the vertices of a k-dominating set. After a vertex is attacked, guards “defend” by each move up to distance k to form a k-dominating set containing the attacked vertex. The eternal k-domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the k = 1 situation and we introduce eternal k-domination for k > 1. Determining if a given set is an eternal k-domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we utilize decomposition arguments to bound the eternal k-domination numbers, and solve the problem entirely in the case of perfect m-ary trees.

In the Firefighter problem, a fire breaks out at a vertex of a graph and at each subsequent time step, the firefighter chooses a vertex to protect and then the fire spreads from each burned vertex to every unprotected neighbour. The problem can be thought of as a simplified model for the spread of gossip or disease in a network. We introduce a new two-player variation called the Pyro game, in which at each step, the fire spreads from one burned vertex to all unprotected neighbours of that vertex. The fire is no longer automated and aims to maximize the number of burned vertices. We show, that unlike the Firefighter problem, one firefighter can contain a fire on the Cartesian grid in the Pyro game. We also study both the Pyro Game and the Firefighter Problem on the infinite strong grid and the complexity of the Pyro game.

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.

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 disprove a conjecture proposed in [Gaspers et al., Discrete Applied Mathematics, 2010] and provide a new upper bound for the minimum number of brushes required to continually parallel clean a clique.

The vertex-edge domination number of a graph, γve(G), is defined to be the cardinality of a smallest set D such that there exists a vertex cover C of G such that each vertex in C is dominated by a vertex in D. This is motivated by the problem of determining how many guards are needed in a graph so that a searchlight can be shone down each edge by a guard either incident to that edge or at most distance one from a vertex incident to the edge. Our main result is that for any cubic graph G with n vertices, γve(G) ≤ 9n/26. We also show that it is NP-hard to decide if γve(G) = γ(G) for bipartite graph G.

We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if 2|E(G)| chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research.

We consider the “all guards move” model for the eternal dominating set problem. A set of guards form a dominating set on a graph and at the beginning of each round, a vertex not in the dominating set is attacked. To defend against the attack, the guards move (each guard either passes or moves to a neighboring vertex) to form a dominating set that includes the attacked vertex. The minimum number of guards required to defend against any sequence of attacks is the “eternal domination number” of the graph. In 2005, it was conjectured [Goddard et al. (J. Combin. Math. Combin. Comput. 52:169–180, 2005)] there would be no advantage to allow multiple guards to occupy the same vertex during a round. We show this is, in fact, false. We also describe algorithms to determine the eternal domination number for both models for eternal domination and examine the related combinatorial game, which makes use of the reduced canonical form of games.

A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction number is determined exactly for $m \times n$ grids with $m \leq 4$ and upper bounds are given for all $n \geq m \geq 8$.

The domination number for grid graphs has been a long studied problem; the first results appeared over thirty years ago [Jacobson 1984] and the final results appeared in 2013 [Goncalves 2013]. Grid graphs are a natural class of graphs to consider for the eternal dominating set problem as the domination number forms a lower bound for the eternal domination number. The 3 x n grid has been considered in several papers, and the difference between the upper and lower bounds for the eternal domination number in the all-guards move model has been reduced to a linear function of n. In this short paper, we provide an upper bound for the eternal domination number which exceeds the lower bound by at most 3.