The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) =...

Let G = (V, E) be a simple graph with vertex set V and edge set E. A function f from V to a set {-1, 1} is said to be a nonnegative signed dominating function (NNSDF) if the sum of its function values over any closed neighborhood is at least zero. The weight of f is the sum of function values of vertices in V. The nonnegative signed domination number for a graph G equals the minimum weight of a...

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

Given two graphs G1 and G2, the Kronecker product G1 ⊗G2 of G1 and G2 is a graph which has vertex set V (G1⊗G2) = V (G1)×V (G2) and edge set E(G1 ⊗ G2) = {(u1, v1)(u2, v2) : u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. ∗ Research was partially supported by the National Nature Science Foundation of China (No. 11171207) and the Key Programs of Wuxi City College of Vocational Technology (WXCY2012-GZ-007). † Co...

Let (G) be the domination number of graph G, thus a graph G is k -edge-critical if (G) 1⁄4 k ; and for every nonadjacent pair of vertices u and v, (Gþ uv) 1⁄4 k 1. In Chapter 16 of the book ‘‘Domination in Graphs— Advanced Topics,’’ D. Sumner cites a conjecture of E. Wojcicka under the form ‘‘3-connected 4-critical graphs are Hamiltonian and perhaps, in general (i.e., for any k 4), (k 1)-connec...

For a graph property P and a graph G, a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. A P-Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the set of all vertices with label 1 or 2 is a P-set. The P-Roman domination number γPR(G) of G is the minimum of Σv∈V (...

Bipolar intuitionistic fuzzy graphs (BIFG) with perfectly connected domination are introduced in this study. The study also discusses the bipolar antifuzzy (BIAFG) concepts. Additionally, topic of 2-domination BIAFG explored. we developed secure edge dominating set and its number some results derived suitable examples.

• Improved approximation algorithm for Budgeted Connected Dominating Set. Optimal Star Edge-Vertex Domination. We consider the version of classical Set problem (BCDS). Given a graph G and budget k , we seek connected subset at most vertices maximizing number dominated in . improve over previous ( 1 − e ) / 13 state art [Khuller, Purohit, Sarpatwar, SODA 2014 ] by introducing new method performi...