Signed Edge Domination Research Articles

A conjecture on the lower bound of the signed edge domination number of 2-connected graphs

In this short note, we construct an infinite family of counterexamples to a conjecture on the lower bound of the signed edge domination number of 2-connected graphs. We propose two problems in order to revise the original conjecture.

Upper bounds on the signed edge domination number of a graph

A signed edge domination function (or SEDF) of a simple graph G=(V,E) is a function f:E→{1,−1} such that ∑e′∈N[e]f(e′)≥1 holds for each edge e∈E, where N[e] is the set of edges in G that share at least one endpoint with e. Let γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved for the two cases vodd(G)=0 and veven(G)=0, where vodd(G) (resp. veven(G)) is the number of odd (resp. even) vertices in G. This article proves Xu’s conjecture for veven(G)∈{1,2}. We also show that for any simple graph G of order n, γs′(G)≤n+vodd(G)∕2 and γs′(G)≤n−2+veven(G) when veven(G)>0, and thus γs′(G)≤(4n−2)∕3. Our result improves the best current upper bound of γs′(G)≤⌈3n∕2⌉.

Signed Edge Domination Number of Interval Graphs

Let G=(V,E) be an undirected graph with vertex set V and edge set E. The open neighborhood N(e) of an edge e∈E is the set of all edges adjacent to e. The closed neighborhood of e is denoted by N[e] and N[e]=N(e)∪{e}. A function f:E→{1,−1} is said to be a signed edge dominating function (SEDF), if f satisfies the condition ∑e′∈N[e]f(e′)≥1 for every e∈E. The minimum of the values of ∑e∈Ef(e), taken over all signed edge dominating functions f on G, is called the signed edge domination number (SEDN) of G and is denoted by γs′(G). In this paper, an O(n2) time algorithm is designed to compute the signed edge domination number of interval graphs.

On the trees with same signed edge and signed star domination numbers

ABSTRACTLet and be the signed edge domination number and signed star domination number of a graph G, respectively. In this paper, we prove that for a tree T of order , if then . Moreover, we prove that for any integer , there exists a tree on n=5r+3 vertices such that .

Signed Edge Domination on Rooted Product Graph

Signed Edge Domination on Rooted Product Graph

On the complexity of variations of mixed domination on graphs

This paper is motivated by the concept of the mixed domination problem on graphs and dedicated to the complexity of variations of the problem such as the signed mixed domination, signed edge domination, minus mixed domination, and minus edge domination problems. In this paper, we present NP-completeness and MAX SNP-hardness results for some of them, and present a general framework of solving these problems and various dominating and covering problems for trees in linear time.

On the signed edge domination number of graphs

Let γ s ′ ( G ) be the signed edge domination number of G . In 2006, Xu conjectured that: for any 2-connected graph G of order n ( n ≥ 2 ) , γ s ′ ( G ) ≥ 1 . In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m , there exists an m -connected graph G such that γ s ′ ( G ) ≤ − m 6 | V ( G ) | . Also for every two natural numbers m and n , we determine γ s ′ ( K m , n ) , where K m , n is the complete bipartite graph with part sizes m and n .

A comment to: Two classes of edge domination in graphs

Let G be a graph of order n and γ s ′ ( G ) denote the signed edge domination number of G . In [B. Xu, Two classes of edge domination in graphs, Discrete Appl. Math. 154 (2006) 1541–1546] it was proved that for any graph G of order n , γ s ′ ( G ) ≤ ⌊ 11 6 n − 1 ⌋ . But the method given in the proof is not correct. In this paper we give an example for which the method of proof given in [1] does not work.

On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4

Let G = ( V , E ) be a simple graph. For an edge e of G , the closed edge-neighbourhood of e is the set N [ e ] = { e ′ ∈ E | e ′ is adjacent to e } ∪ { e } . A function f : E → { 1 , − 1 } is called a signed edge domination function (SEDF) of G if ∑ e ′ ∈ N [ e ] f ( e ′ ) ≥ 1 for every edge e of G . The signed edge domination number of G is defined as γ s ′ ( G ) = min { ∑ e ∈ E f ( e ) | f is an SEDF of G } . In this paper, we characterize all trees T with signed edge domination numbers 1, 2, 3, or 4.

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If $$\sum_{x\in N[e]}f(x) \geq 1$$ for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γs′(G) of G is defined as $$\gamma_s^\prime(G) = {\text{min}}\{\sum_{e\in E(G)}f(e)\mid f \,\text{is an SEDF of} G\}$$. Recently, Xu proved that γs′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γs′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γs′(G) = 1 − k, 2 − k.

A note on the signed edge domination number in graphs

Let G = ( V ( G ) , E ( G ) ) be a graph. A function f : E ( G ) → { + 1 , − 1 } is called the signed edge domination function (SEDF) of G if ∑ e ′ ∈ N [ e ] f ( e ′ ) ≥ 1 for every e ∈ E ( G ) . The signed edge domination number of G is defined as γ s ′ ( G ) = min { ∑ e ∈ E ( G ) f ( e ) | f is a SEDF of G }. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that γ s ′ ( G ) ≤ ⌊ 11 6 n − 1 ⌋ for any graph G of order n ( n ≥ 4 ) . In the article, he conjectured that: For any 2-connected graph G of order n ( n ≥ 2 ) , γ s ′ ( G ) ≥ 1 . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k -connected graphs G m , k with γ s ′ ( G m , k ) = − k ( m − 1 ) ( k m + k + 1 ) 2 ( k ≥ 2 , m ≥ 1 ) .

Two classes of edge domination in graphs

Let γ s ′ ( G ) ( γ l ′ ( G ) , resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179–189]. In this paper, we prove mainly that γ s ′ ( G ) ⩽ ⌊ 11 6 n - 1 ⌋ and γ l ′ ( G ) ⩽ 2 n - 4 hold for any graph G of order n ( n ⩾ 4 ) , and pose several open problems and conjectures.

On edge domination numbers of graphs

Let γ s ′ ( G ) and γ ss ′ ( G ) be the signed edge domination number and signed star domination number of G , respectively. We prove that 2 n - 4 ⩾ γ ss ′ ( G ) ⩾ γ s ′ ( G ) ⩾ n - m holds for all graphs G without isolated vertices, where n = | V ( G ) | ⩾ 4 and m = | E ( G ) | , and pose some problems and conjectures.

Edge Domination in Graphs of Cubes

The signed edge domination number and the signed total edge domination number of a graph are considered; they are variants of the domination number and the total domination number. Some upper bounds for them are found in the case of the n-dimensional cube Q n.

On signed edge domination numbers of trees

On signed edge domination numbers of trees

On signed edge domination numbers of graphs

Given a graph G=(V,E), if e=uv∈E, then the closed edge-neighbourhood of e is denoted by N[e]={u′v′∈E|u′=u or v′=v}. A function f : E→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑ e′∈N[e] f(e′)⩾1 for every e∈E. The signed edge domination number γ s′(G) of G is defined as γ s′(G)= min{∑ e∈E f(e) | f is an SEDF of G}. Let Ψ(m)= min{γ s′(H)|H is a graph with |E(H)|=m}. In this paper, we determine the exact value of Ψ(m) for each positive integer m. That is: Ψ(m)=2 1 3 24m+25 +6m+5 6 −m, where ⌈x⌉ denotes the ceiling of x. In addition, we also characterize all connected graphs G with γ s′(G)=|E(G)| .