Characterizations and algorithms for generalized Cops and Robbers games

We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the game enters into a fixed set of final positions, then the Pursuer wins; otherwise, the Evader wins. A relational characterization of the games where the Pursuer wins is provided. A precise formula is given for the length of the game, along with an algorithm for computing if the Pursuer has a winning strategy whose complexity is a function of the parameters of the game. For games where the position of one player does not affect the available moves of he other, a vertex elimination ordering characterization, analogous to a cop-win ordering, is given for when the Pursuer has a winning strategy

A Study of kk-dipath Colourings of Oriented Graphs

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.Comment: 14 page

Graph homomorphisms with infinite targets

AbstractLet H be a fixed graph whose vertices are called colours. Informally, an H-colouring of a graph G is an assignment of these colours to the vertices of G such that adjacent vertices receive adjacent colours. We introduce a new tool for proving NP-completeness of H-colouring problems, which unifies all methods used previously. As an application we extend, to infinite graphs of bounded degree, the theorem of Hell and Nešetřil that classifies finite H-colouring problems by complexity

Building blocks for the variety of absolute retracts

AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1

Characterizations of k-copwin graphs

AbstractWe give two characterizations of the graphs on which k cops have a winning strategy in the game of Cops and Robber. One of these is in terms of an order relation, and one is in terms of a vertex ordering. Both generalize characterizations known for the case k=1

Oriented Colourings of Graphs with Maximum Degree Three and Four

We show that any orientation of a graph with maximum degree three has an oriented 9-colouring, and that any orientation of a graph with maximum degree four has an oriented 69-colouring. These results improve the best known upper bounds of 11 and 80, respectively

2-limited broadcast domination in grid graphs

We establish upper and lower bounds for the 2-limited broadcast domination number of various grid graphs, in particular the Cartesian product of two paths, a path and a cycle, and two cycles. The upper bounds are derived by explicit constructions. The lower bounds are obtained via linear programming duality by finding lower bounds for the fractional 2-limited multipacking numbers of these graphs

Colourings of (m,n)(m, n)-coloured mixed graphs

A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \geq 0$ colours, and each arc is assigned one of $n \geq 0$ colours. Oriented graphs are $(0, 1)$-coloured mixed graphs, and 2-edge-coloured graphs are $(2, 0)$-coloured mixed graphs. We show that results of Sopena for vertex colourings of oriented graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs and 2-edge-coloured graphs, are special cases of results about vertex colourings of $(m, n)$-coloured mixed graphs. Both of these can be regarded as a version of Brooks' Theorem.Comment: 7 pages, no figure

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete

Bounds for the mm-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The $m$-eternaldomination number, \edom(G), of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence. We show that if $G$ is a connected graph with minimum degree at least~$2$ and of order~$n \ge 5$, then \edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor, and this bound is tight. We also prove that if $G$ is a cubic bipartite graph of order~$n$, then \edom(G) \le \frac{7n}{16}