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...

‎For any integer $kgeq 1$‎, ‎a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-‎tuple total dominating set of $G$ if any vertex‎ ‎of $G$ is adjacent to at least $k$ vertices in $S$‎, ‎and any vertex‎ ‎of $V-S$ is adjacent to at least $k$ vertices in $V-S$‎. ‎The minimum number of vertices of such a set‎ ‎in $G$ we call the $k$-tuple total restrained domination number of $G$‎. ‎The maximum num...

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, ...