A signed bipartite graph G(U, V ) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V ) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V ) is called its signed degree set. In this paper, we prove that every set ...

A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of ...

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E′ ⊆ E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of E′ then E′ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to de...

Abstract— Let G be a bipartite graph. A X-dominating set D of X of G is a strong nonsplit Xdominating set of G if every vertex in X-D is X-adjacent to all other vertices in X-D. The strong nonsplit X-domination number of a graph G, denoted by ) (G snsX γ is the minimum cardinality of a strong nonsplit X-dominating set. We find the bounds for strong nonsplit X-dominating set and give its biparti...

We prove that every (claw, net)-free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present a linear time algorithm which, for a given (claw, net)-free graph, 3nds either a dominating pair or an induced doubly dominating cycle. We show also how one can use structural properties of (claw, net)-free graphs to solve e4ciently the domination, indep...

A set of vertices in a graph is perfect dominating if every vertex outside the set is adjacent to exactly one vertex in the set, and is neighborhood connected if the subgraph induced by its open neighborhood is connected. In any graph the full set of vertices is perfect dominating, and in every connected graph the full set of vertices is neighborhood connected. It is shown that (i) in a connect...

Let $D$ be a finite and simple digraph with vertex set $V(D)$‎.‎A signed total Roman $k$-dominating function (STR$k$DF) on‎‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin g...

We first consider some problems related to the maximum number of dominating (or strong dominating) sets in a regular graph. Our techniques, centered around Shearer’s entropy lemma, extend to a reasonably broad class of graph parameters enumerating vertex colorings that satisfy conditions on the multiset of colors appearing in neighborhoods (either open or closed). Dominating sets and strong dom...

The line graph of an edge-signed graph carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized edge-signed graphs whose line graphs are consistent. Their proof applies Hoede’s relatively difficult characterization of consistent vertex-signed graphs. We give a different, constr...