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 eE(G). The signed edge domination number of G is defined as 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 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), . 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 .     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

On edge domination numbers of graphs

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

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

A comment to: Two classes of edge domination in graphs

Let G be a graph of order n and 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, . 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.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

Two classes of edge domination in graphs

Let (, 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 and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

Upper signed domination number

Let G=(V,E) be a graph. A signed dominating function on G is a function f:V→{-1,1} such that for each v∈V, where N[v] is the closed neighborhood of v. The weight of a signed dominating function f is . A signed dominating function f is minimal if there exists no signed dominating function g such that g≠f and g(v)?f(v) for each v∈V. The upper signed domination number of a graph G, denoted by Γ_s(G), equals the maximum weight of a minimal signed dominating function of G. In this paper, we establish an tight upper bound for Γ_s(G) in terms of minimum degree and maximum degree. Our result is a generalization of those for regular graphs and nearly regular graphs obtained in [O. Favaron, Signed domination in regular graphs, Discrete Math. 158 (1996) 287-293] and [C.X. Wang, J.Z. Mao, Some more remarks on domination in cubic graphs, Discrete Math. 237 (2001) 193-197], respectively.     

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood N_G[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 for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ_s^′(G) of G is defined as . Recently, Xu proved that γ_s^′(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 γ_s^′(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 γ_s^′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar.     

Edge irregular total labellings for graphs of linear size

As an edge variant of the well-known irregularity strength of a graph G=(V,E) we investigate edge irregular total labellings, i.e. functions f:V∪E→{1,2,…,k} such that f(u)+f(uv)+f(v)≠f(u^′)+f(u^′v^′)+f(v^′) for every pair of different edges uv,u^′v^′∈E. The smallest possible k is the total edge irregularity strength of G. Confirming a conjecture by Ivan?o and Jendrol’ for a large class of graphs we prove that the natural lower bound is tight for every graph of order n, size m and maximum degree Δ with m>111000Δ. This also implies that the probability that a random graph from G(n,p(n)) satisfies the Ivan?o-Jendrol’ Conjecture tends to 1 as n→∞ for all functions p∈[0,1]^N. Furthermore, we prove that is an upper bound for every graph G of order n and size m≥3 whose edges are not all incident to a single vertex.     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.     

On global domination critical graphs

A dominating set of a graph G=(V,E) is a subset S⊆V such that every vertex not in S is adjacent to at least one vertex of S. The domination number of G is the cardinality of a smallest dominating set. The global domination number, γ_g(G), is the cardinality of a smallest set S that is simultaneously a dominating set of both G and its complement . Graphs for which γ_g(G−e)>γ_g(G) for all edges e∈E are characterized, as are graphs for which γ_g(G−e)<γ_g(G) for all edges e∈E whenever is disconnected. Progress is reported in the latter case when is connected.     

A better approximation algorithm for the budget prize collecting tree problem

Given an undirected graph G=(V,E), an edge cost c(e)?0 for each edge e∈E, a vertex prize p(v)?0 for each vertex v∈V, and an edge budget B. The BUDGET PRIZE COLLECTING TREE PROBLEM is to find a subtree T′=(V′,E′) that maximizes , subject to . We present a (4+ε)-approximation algorithm.     

Rainbow domination on trees

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,…,k} such that for any vertex v with f(v)=0? we have ∪_u∈NG(v)f(u)={1,2,…,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G, that is the minimum value of ∑_v∈V(G)|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that .     

Total minus domination in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by , of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound on of k-partite graphs is given.     

Broadcasts in graphs

We say that a function f:V→{0,1,…,diam(G)} is a broadcast if for every vertex v∈V, f(v)?e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value . In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.     

Efficient algorithms for Roman domination on some classes of graphs

A Roman dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex x with f(x)=0 is adjacent to at least one vertex y with f(y)=2. The weight of a Roman dominating function is defined to be f(V)=∑_x∈Vf(x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer an open question mentioned in [E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11-22] by showing that the Roman domination number of an interval graph can be computed in linear time. We then show that the Roman domination number of a cograph (and a graph with bounded cliquewidth) can be computed in linear time. As a by-product, we give a characterization of Roman cographs. It leads to a linear-time algorithm for recognizing Roman cographs. Finally, we show that there are polynomial-time algorithms for computing the Roman domination numbers of -free graphs and graphs with a d-octopus.     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and