Vertex-edge Domination Research Articles

Algorithmic Aspects of Vertex-edge Domination in Some Graphs

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set \(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

Connected Vertex-Edge Dominating Sets and Connected Vertex-Edge Domination Polynomials of Friendship F_n

Let be a simple connected graph of order Let be the family of connected vertex-edge dominating sets in with cardinality .The polynomial is called the connected vertex - edge domination polynomial of G, where is the number of connected vertex - edge dominating sets of G. In this paper, we study some properties of connected vertex-edge domination polynomials of the Friendship graph . We obtain a recursive formula for . Using this recursive formula, we construct the connected vertex - edge domination polynomial of , where is the number of the connected vertex - edge dominating sets of of cardinality and some properties of this polynomial have been studied.

Algorithmic Aspects of Total Vertex-Edge Domination in Graphs

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

Graphs with unique minimum vertex-edge dominating sets

A vertex u of a graph G = (V, E), ve-dominates every edge incident to u, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve-dominated by at least one vertex of S. The vertex-edge domination number is the minimum cardinality of a ved-set in G. In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.

Vertex-edge domination in unit disk graphs

Let G = ( V , E ) be a simple undirected graph. A set D ⊆ V is called a vertex-edge dominating set of G if for each edge e = u v ∈ E , either u or v is in D or one vertex from their neighbor is in D . Simply, a vertex v ∈ V , vertex-edge dominates every edge u v , as well as every edge adjacent to these edges. The objective of the vertex-edge domination problem is to find a minimum size vertex-edge dominating set of G . Herein, we study the vertex-edge domination problem in unit disk graphs and prove that the decision version of the problem belongs to the NP-complete class. We also show that the problem admits a simple polynomial-time 4-factor approximation algorithm and a polynomial-time approximation scheme (PTAS) in unit disk graphs.

Total outer connected vertex-edge domination

A vertex [Formula: see text] of a graph is said to [Formula: see text]-[Formula: see text] dominate every edge incident with [Formula: see text], as well as every edge incident to vertices adjacent to [Formula: see text]. A subset [Formula: see text] is a total outer connected vertex-edge dominating set of a graph [Formula: see text] if every edge in [Formula: see text] is [Formula: see text]-[Formula: see text] dominated by a vertex in [Formula: see text], the subgraph induced by [Formula: see text] has no isolated vertices and the subgraph induced by [Formula: see text] is connected. We initiate the study of total outer connected vertex-edge domination in graphs. We show that the decision problem for total outer-connected vertex-edge domination problem is [Formula: see text]-Complete even for bipartite graphs. We prove that for every tree of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text] and characterize the trees attaining the lower bound. We also study the effect of edge removal, edge addition and edge subdivision on total outer connected vertex-edge domination number of a graph.

Results on vertex-edge and independent vertex-edge domination

Results on vertex-edge and independent vertex-edge domination

Total vertex-edge domination in graphs: Complexity and algorithms

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

Connected Vertex-Edge Dominating Sets and Connected Vertex-Edge Domination Polynomials of Gem Graph Gn

Let G be a simple connected graph of order n. Let D c ν e (G, i) be the family of connected vertex-edge dominating sets of G with cardinality i. The polynomial D c ν e (G, x) = is called the connected vertex-edge domination polynomial of G where d c ν e (G, i) is the number of vertex edge dominating sets of G. In this paper, we study some properties of connected vertex-edge domination polynomials of the Gem graph G n . Also we obtain some properties of D c ν e (G n , x) and their coefficients. Also, we find the recursive formula, to derive the connected vertex-edge dominating sets of the Gem graph G n .

Connected vertex–Edge dominating sets and connected vertex–Edge domination polynomials of triangular ladder

Connected vertex–Edge dominating sets and connected vertex–Edge domination polynomials of triangular ladder

Double vertex-edge domination in graphs: complexity and algorithms

In a graph $$G=(V,E)$$ , a vertex $$v\in V$$ is said to ve-dominate the edges incident on v as well as the edges adjacent to these incident edges on v. A set $$D\subseteq V$$ is called a double vertex-edge dominating set if every edge of the graph is ve-dominated by at least two vertices of D. Given a graph G, the double vertex-edge dominating problem, namely Min-DVEDS is to find a minimum double vertex-edge dominating set of G. In this paper, we show that the decision version of Min-DVEDS is NP-complete for chordal graphs. We present a linear time algorithm to find a minimum double vertex-edge dominating set in proper interval graphs. We also show that for a graph having n vertices, Min-DVEDS cannot be approximated within $$(1 -\varepsilon ) \ln n$$ for any $$\varepsilon > 0$$ unless NP $$\subseteq $$ DTIME( $$n^{O(\log \log n)}$$ ). On positive side, we show that Min-DVEDS can be approximated by a factor of $$O(\ln \varDelta )$$ . Finally, we show that Min-DVEDS is APX-complete for graphs with maximum degree 5.

A lower bound on the total vertex-edge domination number of a tree

A vertex [Formula: see text] of a graph [Formula: see text] is said to vertex-edge dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A subset [Formula: see text] is a vertex-edge dominating set (ve-dominating set) if every edge of [Formula: see text] is vertex-edge dominated by at least one vertex of [Formula: see text]. A vertex-edge dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of [Formula: see text], denoted by [Formula: see text], is called the total vertex-edge domination number of [Formula: see text]. In this paper, we prove that for every nontrivial tree of order [Formula: see text], with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize extremal trees attaining the lower bound.

The outer-connected vertex edge domination number in Cartesian product graphs

Let G be a simple, finite and connected graph. An outer-connected vertex edge dominating set, abbreviated OCVEDS of G is a subset D of vertices of G such that D is a vertex edge dominating set and the graph G\\D is connected. The outer-connected vertex edge domination number of G is a minimum cardinality of a OCVEDS of G and is denoted by . This paper focuses on the OCVEDS in Cartesian product of graphs.

On Two Open Problems on Double Vertex-Edge Domination in Graphs

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

The outer-connected vertex edge domination number in Lexicographic product graphs

The outer-connected vertex edge domination number in Lexicographic product graphs

Total domination and vertex-edge domination in tres

A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−l+1)/2 ≤ γve(T) ≤(γt(T)+l−1)/2 and characterize trees attaining each of these bounds.

Influence of the Edge Removal, Edge Addition and Edge Subdivision on the Double Vertex–Edge Domination Number of a Graph

A vertex v of a graph $$G=(V,E)$$ is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set $$S \subseteq V$$ is a vertex–edge dominating set (double vertex–edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a vertex–edge dominating set (double vertex–edge dominating set, respectively) of G is the vertex–edge domination number $$\gamma _{ve}(G)$$ (the double vertex–edge domination number $$\gamma _{dve}(G)$$ , respectively). The influence of edge removal, edge addition and edge subdivision on the double vertex–edge domination number of a graph are investigated in this paper.

Double vertex-edge domination

A vertex [Formula: see text] of a graph [Formula: see text] is said to [Formula: see text]-dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A set [Formula: see text] is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of [Formula: see text] is [Formula: see text]-dominated by at least one vertex (at least two vertices) of [Formula: see text] The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of [Formula: see text] is the vertex-edge domination number [Formula: see text] (the double vertex-edge domination number [Formula: see text], respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number [Formula: see text] for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs [Formula: see text] [Formula: see text] and we characterize the trees [Formula: see text] with [Formula: see text] or [Formula: see text] Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order [Formula: see text] the number of leaves and support vertices, and we characterize the trees attaining the lower bound.

Total vertex-edge domination

ABSTRACTA vertex v of a graph is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domination number A ve-dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total ve-dominating set of G is the total vertex-edge domination number In this paper we initiate the study of total vertex-edge domination. We show that determining the number for bipartite graphs is NP-complete. Then we show that if T is a tree different from a star with order n, ℓ leaves and s support vertices, then Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that and we provide a characterization of all trees T with .

A note on independent vertex–edge domination in graphs

The independent vertex–edge domination number and the upper non-enclaving number of a graph G are denoted by ive(G) and Ψ(G), respectively. Boutrig et al. posed the following question: Let G be a connected graph with order n. Is Ψ(G)+ive(G)≤n?In this paper, we provide an infinite family of counterexamples. A new relationship between Ψ(G) and ive(G) is established. Furthermore, if G is a connected cubic graph, we answer this question in the affirmative.