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...

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

This paper is motivated by the concept of the signed k-domination problem and dedicated to the complexity of the problem on graphs. For any fixed nonnegative integer k, we show that the signed k-domination problem is NP-complete for doubly chordal graphs. For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination problem can be solved in polynomial time. We...

Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied Hamiltonian-related properties of supergrid and their subclasses graphs. this paper, propose a generalized graph class called extended Extended include grid graphs, diagonal triangular as study problems domination independent on A dominating set is subset vertices it, such t...

We initiate the study of total outer-independent domination in graphs. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the ...

For a graph G, let f : V (G) → P({1, 2, . . . , k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, . . . , k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight, w(f), of a kRDF f is defined as w(f) = ∑ v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by ...

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u...

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...