The copnumber for lexicographic products and sums of graphs

For the lexicographic product G • H of two graphs G and H so that G is connected, we prove that if the copnumber c(G) of G is greater than or equal to 2, then c(G • H) = c(G). Moreover, if c(G) = c(H) = 1, then c(G • H) = 1. If c(G) = 1, G has more than one vertex, and c(H) ≥ 2, then c(G • H) = 2. We also provide the copnumber for general lexicographic sums. The game of cops and robbers on a graph, conceived originally in [4] and recently described comprehensively in [2], is played according to the following rules: Vertices v1, . . . , vn in a graph are chosen as the initial positions for cops C1, . . . , Cn. A vertex w is then chosen for a robber R. At the start, as well as throughout the game, multiple cops can occupy the same vertex. The cops’ objective is to catch the robber by placing a cop on the same vertex with the robber. The robber’s objective is to prevent this from happening. Both sides know the position of all cops and of the robber at all times. Each side alternately takes turns, starting with the cop. A cop move consists of each of the cops either staying at their current vertex or moving to an adjacent vertex. In a robber move, the robber either stays at their current vertex or moves to an adjacent vertex. The smallest number of cops needed to capture the robber in a given graph G is called the graph’s copnumber c(G). A graph with c(G) = 1 is called cop-win. It is customary to label the vertex at which the cop Ck is located and the vertex at which the robber R is located by Ck and R respectively which we will use throughout the paper. Moreover, since neither the cops nor the robber can leave the component in which each of them started the game, the game is typically assumed to be played on a connected graph. Trivially, because the robber is placed after the cops are placed, the copnumber of a disconnected graph is the sum of the copnumbers of its components. See [2, Section 4.2] for a survey of the copnumbers for several types of products for graphs. Aside from results on the cartesian product and the strong Received by the editors October 11, 2012, and in revised form December 4, 2013. 2010 Mathematics Subject Classification. 05C57, 91A43, 05C75, 05C85.