Prabhakar and Hilfer-Prabhakar Fractional Derivatives in the Setting of $\Psi$-Fractional Calculus and Its Applications
Download PDF
Authors: S. K. MAGAR, P. V. DOLE AND K. P. GHADLE
DOI: 10.46793/KgJMat2105.685K
Abstract:
A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.
Keywords:
Vertex-edge roman dominating set, edge dominating set, trees.
References:
[1] R. Boutrig, M. Chellali, T. W. Haynes and S. T. Hedetniemi, Vertex-edge domination in graphs, Aequationes Math. 90 (2016), 355–366.
[2] M. Chellali, T. W. Haynes and S. T. Hedetniemi, Bounds on weak Roman and 2-rainbow domination numbers, Discrete Appl. Math. 178 (2014), 27–32.
[3] M. Chellali and N. Jafari Rad, Trees with unique Roman dominating functions of minimum weight, Discrete Math. Algorithms Appl. 6 (2014), Paper ID 1450038.
[4] E. J. Cockayne, P. A. Dreyer, S. M. Hedetniemi and S. T. Hedetniemi, Roman domination in graphs, Discrete Math. 78 (2004), 11–22.
[5] B. Krishnakumari, Y. B. Venkatakrishnan and M. Krzywkowski, Bounds on the vertex-edge domination number of a tree, C. R. Math. Acad. Sci. Paris 352 (2014), 363–366.
[6] J. R. Lewis, S. T. Hedetniemi, T. W. Haynes and G. H. Fricke, Vertex-edge domination, Util. Math. 81 (2010), 193–213.
[7] S. Mitchell and S. T. Hedetniemi, Edge domination in trees, Congr. Numer. 19 (1977), 489–509.
[8] K. W. Peters, Theoretical and algorithmic results on domination and connectivity, Ph.D. Thesis, Clemson University, 1986.
[9] E. N. Satheesh, Some variations of domination and applications, Ph.D. Thesis, Mahatma Gandhi University, 2014.
Login:
