The vertices and edges of a graph G are called the elements of G. A set X of elements in G is an entire dominating set if every element not in X is an entire dominating set if every element not in X is either adjacent or incident to at least one element in X. An entire dominatin...
We define the edge lift of uv off x as the process of removing edges ux and vx while adding the edge uv to G. In this paper, we investigate the effect that edge lifting has on the total domination number of a graph. Among other results, we show that there are no trees for which ...
The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this ...
A graph \\\(G\\\) is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph \\\(G\\\) of order \\\(n\\\) is at most \\\(\\\lfloor n^2/4 floor...
The total dominator chromatic (TDC) number x_d~t(G) of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G_1 and G_2 is denoted by G_1 * G_2 and is the graph obtained by taking one copy of G_1 and | V(G_1...
We initiate the study of signed edge majority total domination in graphs. The open neighborhood N_G(e) of an edge e in a graph G is the set consisting of all edges having a common vertex with e. Let f be a function on E(G), the edge set of G, into the set {-1, 1}. If ...
Let G = (V (G), E(G)) be a graph. An mmatchings of G is a set of edges of size m in which any two edges are mutually independent. Denote by z(G, m) the number of mmatchings of G. Let z(G) be the total number of matchings in G, namely z(G... SW Yuwen Chen 被引...
In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r r -gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G G is the minimum number of vertices (edges) ...
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A graph is said to be threshold if there exist real numbers a i associated with its vertices i and a real number b such that ∑ i S a i b holds if and only if S is a stable set of vertices. A vertex of a graph is said to cover itself, its neighboring vertices, and the incid...