Double Edge-Vertex Domination on Middle and Splitting Graphs of Path and Cycle

Year 2021, Volume: 4 Issue: 4, 221 - 231, 01.12.2021

Banu Aylı , Elgin Kılıc

https://doi.org/10.33401/fujma.916263

Abstract

An edge $e=uv$ of graph $G=(V,E)$ is said to be edge-vertex dominate vertices $u$ and $v$, as well as all vertices adjacent to $u$ and $v$. A set $S \subseteq E $ is a double edge-vertex dominating set if every vertex of $V$ is edge-vertex dominated by at least two edges of $S$. The minimum cardinality of a double edge-vertex dominating set of $G$ is the double edge-vertex domination number and is denoted by $\gamma_{dev}(G)$. In this paper, we present results for middle graphs of path and cycle and some splitting graphs of path and cycle on double edge-vertex domination numbers.

Keywords

Double edge-vertex dominating set , Double vertex-edge dominating set , Edge-vertex dominating set , Vertex-edge dominating set

References

• [1] O. Ore, Theory of graphs, American Mathematical Society, Providence, RI, 30 (1962), 206-212.
• [2] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker Inc., New York, 1998.
• [3] E. Kılıc¸, B. Aylı, Double edge-vertex domination number of graphs, Adv. Math. Mod. Appl., 5(1), (2020), 19-37.
• [4] E. Kılıc¸, B. Aylı, Double edge-vertex domination under some graph operations, J. Mod. Tech. Eng., 5(2), (2020), 175-180.
• [5] V.R. Kulli, The neighborhood graph of a graph, International Journal of Fuzzy Mathematical Archive, 8(2), (2015), 93-99.
• [6] S. Mitchell, S.T. Hedetniemi, Edge domination in trees, Congr. Numer., 19, (1977), 489-509.
• [7] J.W. Peters, Theoretical and algorithmic results on domination and connectivity, Ph.D.Thesis, Clemson University, 1986.
• [8] J.R. Lewis, Edge and edge-vertex domination in graphs, Ph.D. Thesis, Clemson University, 2007.
• [9] B. Krishnakumari, M. Chellali, Y.B. Venkatakrishnan, Double vertex-edge domination, Disc. Math., Algorithms and Applications, 9(4), (2017), Article ID 1750045, doi:10.1142/S1793830917500458.
• [10] E. Sampathkumar, H.B. Walikar, On splitting graph of a graph, J. Karnatak Univ. Sci., 25(13), (1980), 13-16.
• [11] I. Hamada, T. Yoshimura, Traversability and connectivity of the middle graph of a graph, Disc. Math., 14(3), (1976), 247-256.