Generous Roman Domination Subdivision Number in Graph

Let be an integer and G a simple graph with vertex set V(G). Let f be a function that assigns labels from the set to the vertices of G. For a vertex the active neighbourhood AN(v) of v is the set of all vertices w adjacent to v such that A [k]-Roman dominating function (or [k]-RDF for short) is a function satisfying the condition that for any vertex with f(v) < k, The weight of a [k]-RDF is and the [k]-Roman domination number of G is the minimum weight of an [k]-RDF on G. In this paper we shall be interested in the study of the [k]-Roman domination subdivision number sd of G defined as the minimum number of edges that must be subdivided, each once, in order to increase the [k]-Roman domination number. We first show that the decision problem associated with sd is NP-hard. Then various properties and bounds are established.

A note on the saturation number of the family of [formula omitted]-connected graphs

Let G=V,E be a simple graph. A subset D⊆V is a dominating set if every vertex not in D is adjacent to a vertex in D. The domination number of G, denoted by γG, is the smallest cardinality of a dominating set of G. The domination subdivision number sdγG of G is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) in order to increase the domination number. In 2000, Haynes et al. showed that sdγG≤dGv+dGv−1 for any edge uv∈EG with dGu≥2 and dGv≥2 where G is a connected graph with order no less than 3. In this paper, we improve the above bound to sdγG≤dGu+dGv−NGu∩NGv−1, and furthermore, we show the decision problem for determining whether sdγG=1 is NP-hard. Moreover, we show some bounds or exact values for domination subdivision numbers of some graphs.

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex u for which [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is undefended if it is not adjacent to a vertex with [Formula: see text]. The function [Formula: see text] is a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text] such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] if [Formula: see text] has no undefended vertex. The Roman domination subdivision number of a graph [Formula: see text] is the minimum number of edges that must be subdivided in order to increase the Roman domination number of [Formula: see text]. We introduce the concept of weak Roman subdivision number of a graph [Formula: see text], denoted by [Formula: see text] as the minimum number of edges that must be subdivided in order to increase the weak Roman domination number of [Formula: see text]. In this paper, we determine the exact values of the weak Roman subdivision number for paths, cycles and complete bipartite graphs. We obtain bounds for the weak Roman subdivision number of a graph and characterize the extremal graphs.

NP-hardness of multiple bondage in graphs

This paper discusses about connectivity algorithm in simple graphs. The algorithm has three stages, namely, adjacent vertices search, adjacent vertices investigation and component investigation. The aim of connectivity algorithm is to check whether a graph is connected or disconnected with the scope set. There are three sets which are used in this algorithm, namely, vertex, edge and adjacent vertices sets. The simple graph is connected if the graph has only one component or if the number of adjacent vertices elements is equal to the number of vertices. Otherwise, if the simple graph has more than one component or if the number of adjacent vertices elements is not equal to the number of vertices then the algorithm will conclude that the graph is disconnected.

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. We give a counterexample to this conjecture. On the other hand, we show that sdγ(G) ≤ γ(G)+1 for any graph G without isolated vertices, and give constant upper bounds on sdγ(G) for several families of graphs.

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

Let $G = (V, E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ with $f(v) = 2$. The weight of a Roman dominating function is the value $f(G) = \sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. Rad and Volkmann [9] proposed a problem that is to determine the trees $T$ with Roman bondage number $1$. In this paper, we characterize trees with Roman bondage number $1$.

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice Λ by ℓ bonds connecting the same adjacent vertices, thereby yielding the lattice Λ ℓ . This relation is used to calculate the bond percolation threshold on Λ ℓ . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d≥2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N→∞ limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N→∞.

Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wєV-D, N(u)∩D ≠ N(w)∩D, |N(u)∩D| ≠ |N(w)∩D|. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the locating equitable domination number and is denoted by sdγle(G). The independence subdivision number sdβle(G) to equal the minimum number of edges that must be subdivided in order to increase the independence number. In this paper, we establish bounds on sdγle(G) and sdβle(G) for some families of graphs.

A Roman dominating function on a graph G = (V, E) is a function \(f : V \rightarrow \{0, 1, 2\}\) satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value \(w(f) = \sum_{v\in V} f(v)\). The Roman domination number of a graph G, denoted by \(_{\gamma R}(G)\), equals the minimum weight of a Roman dominating function on G. The Roman domination subdivision number \(sd_{\gamma R}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number. In this paper, first we establish upper bounds on the Roman domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that \(1 \leq sd_{\gamma R}(G) \leq 3\). Finally, we show that the Roman domination subdivision number of a graph can be arbitrarily large.

Upper bounds for the Roman domination subdivision number of a graph

A set \(S\) of vertices in a graph \(G = (V,E)\) is a \(2\)-dominating set if every vertex of \(V\setminus S\) is adjacent to at least two vertices of \(S\). The \(2\)-domination number of a graph \(G\), denoted by \(\gamma_2(G)\), is the minimum size of a \(2\)-dominating set of \(G\). The \(2\)-domination subdivision number \(sd_{\gamma_2}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the \(2\)-domination number. The authors have recently proved that for any tree \(T\) of order at least \(3\), \(1 \leq sd_{\gamma_2}(T)\leq 2\). In this paper we provide a constructive characterization of the trees whose \(2\)-domination subdivision number is \(2\).

The domination subdivision number sd∞(G) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number sdfl(G) to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ‚ 2, either G = K1;m and sdfl(G) = m, or 1 • sdfl(G) • 2. We also characterize the graphs G for which sdfl(G) = 2.