A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant kF such that, for every connected F-free graph G, rc(G) ≤ diam...

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we prove several non-trivial upper bounds for rc(G), as well as determine sufficient conditions that gua...

Abstract In this paper, we study the problem of deciding whether total domination number a given graph G can be reduced using exactly one edge contraction (called 1 -Edge Contraction( ? t ) ). We focus on several classes and determine computational complexity problem. By putting together these results, manage to obtain complete dichotomy for H-free graphs.

A cycle C in a graph G is dominating if every edge of G is incident with at least one vertex of C. For a set H of connected graphs, a graph G is said to be H-free if G does not contain any member of H as an induced subgraph. When |H| = 2, H is called a forbidden pair. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominatin...

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that every two distinct vertices of G are connected by k...

OF THE DISSERTATION Applications and Variations of Domination in Graphs by Paul Andrew Dreyer, Jr. Dissertation Director: Fred S. Roberts In a graph G = (V, E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications...