Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs

Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset $S\subseteq V(G)$ is digitally convex if, for every $v\in V(G)$, we have $N[v]\subseteq N[S]$ implies $v\in S$. The number of cyclic binary strings with blocks of length at least $k$ is expressed as a linear recurrence relation for $k\geq 2$. A bijection is established between these cyclic binary strings and the digitally convex sets of the $(k-1)^{th}$ power of a cycle. A closed formula for the number of digitally convex sets of the Cartesian product of two complete graphs is derived. A bijection is established between the digitally convex sets of the Cartesian product of two paths, $P_n \square P_m$, and certain types of $n \times m$ binary arrays.Comment: 16 pages, 3 figures, 1 tabl

Vertices contained in all or in no minimum total dominating set of a tree

AbstractA set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. We characterize the set of vertices of a tree that are contained in all, or in no, minimum total dominating sets of the tree

Block Graphs with Large Paired Domination Multisubdivision Number

The paired domination multisubdivision number of a nonempty graph G, denoted by msdpr(G), is the smallest positive integer k such that there exists an edge which must be subdivided k times to increase the paired domination number of G. It is known that msdpr(G) ≤ 4 for all graphs G. We characterize block graphs with msdpr(G) = 4

Block Graphs with Large Paired Domination Multisubdivision Number

The paired domination multisubdivision number of a nonempty graph G, denoted by msdpr(G), is the smallest positive integer k such that there exists an edge which must be subdivided k times to increase the paired domination number of G. It is known that msdpr(G) ≤ 4 for all graphs G. We characterize block graphs with msdpr(G) = 4

Triangle Decompositions of Planar Graphs

A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1

Triangle Decompositions of Planar Graphs

A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0, 1 and 1/2 . This result provides a characterization of rationally triangle decomposable simple planar graphs. Finally, if G is a multigraph with K4 as underlying graph, we give necessary and sufficient conditions on the multiplicities of its edges for G to be triangle and rationally triangle decomposable