For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

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 graph G is signed if each edge is assigned + or −. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign − if and only if its endpoints are in different parts. The Edwards-Erdös bound states that every graph with n vertices and m edges has a balanced subgraph with at least m 2 +n−1 4 edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max C...

A graph G is signed if each edge is assigned + or −. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign − if and only if its endpoints are in different parts. The Edwards-Erdös bound states that every graph with n vertices and m edges has a balanced subgraph with at least m 2 +n−1 4 edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max C...

For a nonempty graph G = (V, E), a signed edge-domination of G is a function f : E(G) → {1,−1} such that ∑e′∈NG [e] f (e′) ≥ 1 for each e ∈ E(G). The signed edge-domatic number of G is the largest integer d for which there is a set { f1, f2, . . . , fd} of signed edge-dominations of G such that ∑d i=1 fi (e) ≤ 1 for every e ∈ E(G). This paper gives an original study on this concept and determin...

Yannakakis and Gavril showed in [10] that the problem of finding a maximal matching of minimum size (MMM for short), also called Minimum Edge Dominating Set, is NP-hard in bipartite graphs of maximum degree 3 or planar graphs of maximum degree 3. Horton and Kilakos extended this result to planar bipartite graphs and planar cubic graphs [6]. Here, we extend the result of Yannakakis and Gavril in...

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defined by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or o...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (D) with f(v) = ∅ the condition u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on D with t...