The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k ≥ 2, there is no algorithm for Rainbow k-Coloring running in time 2 3/2), unless ETH fails. Motivated by this negative result we consider two paramet...

Given a graphG, the k-dominating graph ofG, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In parti...

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a ra...

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2...

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such ...

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for pro...

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. O...

We investigate in this paper the set of kadditive capacities dominating a given capacity, which we call the k-additive core. We study its structure through achievable families, which play the role of maximal chains in the classical case (k = 1), and show that associated capacities are elements (possibly a vertex) of the k-additive core when the capacity is (k+1)-monotone. As a particular case, ...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

A subset S of nodes in a graph G is k-connected m-dominating set ((k , m)-cds) if the subgraph G[S] induced by and every $$v \in V {\setminus } S$$ has at least m neighbors S. In k -Connected -Dominating Set m)-CDS) problem, goal to find minimum weight (k, m)-cds node-weighted graph. For $$m \ge k$$ we obtain following approximation ratios. unit disk graphs improve ratio $$O(k \ln k)$$ Nutov (I...