Abstract. Let G = (V,E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ0 st(G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on...

A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G...

The in-neighborhood, I(v), of a vertex v in a digraph D=(V, A) is v together with the set of all vertices sending an arc to v, i.e., vertices u such that (u, v) # A. A subset of V is called dominating if it meets I(v) for every v # V. (To avoid confusion, it must be noted that some authors require in the definition meeting every out-neighborhood.) A set of vertices is called independent if no t...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor $D$, while $D$ paired-dominating a~dominating and the subgraph induced by contains perfect matching. graph $D\!P\!D\!P$-graph it pair $(D,P)$ disjoint sets vertices such that $P$ $G$. The study $D\!P\!D\!P$-graphs was initiated Southey Henning (Cent. Eur. J. Math. 8 (2010) 459--467; Comb. Optim. 22 (2...

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...

We show in this paper that the upper minus domination number −(G) of a claw-free cubic graph G is at most 1 2 |V (G)|. © 2006 Published by Elsevier B.V.

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n + O (n/g) < 3n/10 + O (n/g).