For a graph G = (V,E), a set D ⊆ V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semito...

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of...

In this research paper, it was studied about degree of vertices of a semitotal blocks in fuzzy graphs. In process we obtain some interesting results regarding the degree of the vertices in semitotal blocks in fuzzy graphs. We observed that when ‘B’ is a block of a given fuzzy graph G:(V, σ, μ), then degree of the vertex B in semi total block fuzzy graph TSTBF(G) is equal to the sum of the membe...

In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve problems existing in various disciplines physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one which frequently used identify different physicochemical structural properties molecular graphs. Wiener index is first distance-based TI that c...

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E) and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a...

In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.