6156170
On edge domination numbers of graphs
Let @c"s^'(G) and @c"s"s^'(G) be the signed edge domination number and signed star domination number of G, respectively. We prove that 2n-4>[email protected]"s"s^'(G)>[email protected]"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.
39744832
On signed edge domination numbers of graphs
Abstract 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)| .
24660543
A note on the lower bounds of signed domination number of a graph
Abstract Let G = (V, E) be a graph. For a function f : V → {−1, 1}, the weight of f is w(f) = Σv ∈ V f(v). For a vertex v in V, we define f[v] = Σu∈N[v] f(u). A signed dominating function of G is a function f : V → {−1, 1} such that f[v] ⩾ 1 for all v ∈ V. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. A signed dominating function of a weight γs(G) we call a γs-function of G. In this paper, we study the signed domination problem of general graph, and obtain some lower bounds of the signed domination number of a graph, and show that these lower bounds are sharp, and extend a result in Dunbar et al. (1995).
29802636
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) ≤ ⌊11/6n - 1⌋ and γ's(G) ≤ 2n - 4 hold for any graph G of order n (n ≥ 4), and pose several open problems and conjectures.
31788400
Extremal graphs for inequalities involving domination parameters
Abstract A characterization of n-vertex isolate-free connected graphs G whose domination number γ ( G ) satisfies γ(G)=⌊n/2⌋ is obtained. This result enables us to obtain extremal graphs of inequalities which bound the sum of two domination parameters in isolate-free graphs.
20988039
Two conjectures equivalent to the perfect graph conjecture
Abstract In this note, we study the graphs with the property that each of their induced subgraphs has circular clique number (defined in Section 2) the same as its clique number, prove that a graph satisfies such a property if and only if neither itself nor its complement contains an induced subgraph isomorphic to an odd cycle of length at least five, and then give two conjectures which are equivalent to the perfect graph conjecture.
30641096
On the fixed edge of planar graphs with minimum degree five
Abstract An edge e of a finite and simple graph G is called a fixed edge of G if G - e + e ′ ≅ G implies e ′ = e . In this paper, we show that planar graphs with minimum degree 5 contain fixed edges, from which we prove that a class of planar graphs with minimum degree one is edge reconstructible.
28979330
Connected Factors and Spanning Trees in Graphs
Abstract. Let G be a graph, and g, f, f′ be positive integer-valued functions defined on V(G). If an f′-factor of G is a spanning tree, we say that it is f′-tree. In this paper, it is shown that G contains a connected (g, f+f′−1)-factor if G has a (g, f)-factor and an f′-tree.
34666787
Applying Agent into Web Testing and Evolution
Automatic Web testing and evolution is an effective method to ensure the quality of Web applications. In this paper, we apply Agent into this area, i.e. firstly, we explain the necessity and feasibility, and also give the general description of our method; then, we apply Agent into the Web performance testing, Web regression testing and Web usability evolution; conclusion remarks are given at last.
37647794
On 3-colorable plane graphs without 5- and 7-cycles
In this note, it is proved that every plane graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. This improves the result of [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303-311], and offers a partial solution for a conjecture of Borodin and Raspaud [O.V. Borodin, A. Raspaud, A sufficient condition for planar graphs to be 3-colorable, J. Combin. Theory Ser. B 88 (2003) 17-27].