Cops and Robber on some families of oriented graphs

Cops and Robbers on intersection graphs

Cops and Robber is one of the most studied two-player pursuit-evasion games played on graphs, where multiple cops, controlled by one player, pursue a single robber. The cop number of a graph is the minimum number of cops that can ensure the capture of the robber. In directed graphs, two kinds of moves are defined for players: strong move, where a player can move both along and against the orientation of an arc to an adjacent vertex; and weak move, where a player can only move along the orientation of an arc to an out-neighbor. We study three variants of Cops and Robber on oriented graphs: strong cop model, where the cops can make strong moves while the robber can only make weak moves; normal cop model, where both cops and the robber can only make weak moves; and weak cop model, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.

Localization game on geometric and planar graphs

We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number $k$ and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph.

On the Cop Number of a Graph

The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for both relations) the graphs which force bounded cop number. In closing, we bound the cop number in terms of the tree-width.

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

Wall Cops and Robbers is a new turn-based game played on graphs. It is inspired by the games of Cops and Robbers and the Angel Problem. The objective of the game is for the cops to capture the robber by surrounding him with walls. The wall cop number of a graph G, written Wc(G), is the least number of cops it takes to capture the robber in G. The thesis explores these two parameters for various graph classes including trees, hypercubes, grids, and tilings of the plane. We also introduce a variant of the game called Wall Cops and Wall Robbers. In this game, the robber creates a wall on visiting a vertex, disallowing a move to a previously visited vertex.

We study graph searching games where a number of cops try to capture a robber that is hiding in a system of tunnels modelled as a graph. While the current position of the robber is unknown to the cops, each cop can see a certain radius d around his position. For the case d = 1 these games have been studied by Fomin, Kratsch and Müller [7] under the name domination games.We are primarily interested in questions concerning the complexity and monotonicity of these games. We show that dominating games are computationally much harder than standard graph searching games where the cops only see their own vertex and establish strong non-monotonicity results for various notions of monotonicity which arise naturally in the context of domination games. Answering a question of [7], we show that there exists graphs for which the shortest winning strategy for a minimal number of cops must necessarily be of exponential length. On the positive side, we establish tractability results for graph classes of bounded degree.KeywordsDomination NumberWinning StrategyGraph ClassSearch NumberMonotone StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

A k-bisection of a graph is a partition of the vertices in two classes whose cardinalities differ of at most one and such that the subgraphs induced by each class are acyclic with all connected components of order at most k. Esperet, Tarsi and the second author proved in 2017 that every simple cubic graph admits a 3-bisection. Recently, Cui and Liu extended that result to the class of simple subcubic graphs. Their proof is an adaptation of the quite long proof of the cubic case to the subcubic one. Here, we propose an easier proof of a slightly stronger result. Indeed, starting from the result for simple cubic graphs, we prove the existence of a 3-bisection for all cubic graphs (also admitting parallel edges). Then we prove the same result for the larger class of subcubic graphs as an easy corollary.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

A dominating set $$D \subseteq V$$D⊆V in a graph $$G=(V,E)$$G=(V,E) is a subset of vertices such that every vertex in V belongs to D or has at least one neighbour in D. In this paper we deal with the problem of finding an approximation of the dominating set of minimum size, i.e., the approximation of the minimum dominating set problem (MDS) in a distributed setting. A distributed algorithm that runs in a constant number of rounds, independent of the size of the network, is called local. In research on distributed local algorithms it is commonly assumed that each vertex has an unique identifier. However, as was shown by Goos et al., for certain classes of graphs (for example, lift-closed bounded degree graphs) identifiers are unnecessary and only a port numbering is needed. We confirm that the same remains true for the MDS up to a constant factor in the class of planar graphs. Namely, we present a local deterministic 694-approximation algorithm for the MDS in planar graphs in a model with a port numbering only. Moreover, our algorithm uses only short messages, i.e., in each round each node can send only a $$O(\log {|V|})$$O(log|V|)-bit message to each of its neighbours.

When we want to draw a graph to make the information contained in its structure easily accessible, it is highly desirable to have a drawing with as few edge crossings as possible (Purchase et al., 1997; Purchase, 1997). The class of graphs that can be drawn with no crossings at all is the class of planar graphs. Algorithms for drawing planar graphs are the main subject of this chapter.

An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by constructing planar graphs that require four pages in all of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.