The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of 𝒟 is to minimize the total domination number γt(G′) of G′ while 𝒜 tries to maximize it. If both 𝒜 and 𝒟 play according to their optimal strategies, γt(G...
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...
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) ...
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...
In this section, we establish an upper bound on the restrained domination number of a graph with minimum degree at least 2 in terms of its order and maximum degree. We shall prove: Theorem 1 If G is a graph of order n with δ(G)⩾2, then γr(G)⩽n-Δ(G). Proof Let Δ=...
Such a set S is called a locating-total dominating set in G, and the locating-total domination number of G is the minimum cardinality of a locating-total dominating set in G. We obtain new lower and upper bounds on the locating-total domination number of a graph. Interpolation results are...
Total coloring in graph theory refers to assigning colors to both the vertices and edges of a graph such that no two adjacent vertices, no two adjacent edges, and no edge and its incident vertices share the same color.The goal is to find the minimum number of colors required to color the...
A total coloring of a graph $G$ is a coloring of its vertices and edges suchthat no adjacent vertices, edges, and no incident vertices and edges obtain thesame color. An interval total $t$-coloring of a graph $G$ is a total coloringof $G$ with colors $1,\\ldots,t$ such that ...
Füredi, Z.: The maximum number of edges in a minimal graph of diameter 2. J. Graph Theory 16, 81–98 (1992) MathSciNet MATH Google Scholar Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)...
A total-coloring σ of G is called an adjacent vertex distinguishing total-coloring of G if Sσ( u) ≠Sσ( v) for any uv∈E( G),where Sσ( u) denotes the set of colors of edges incident...