Previous |  Up |  Next


domination number; point-set domination number; split domination number; Boolean function graph
For any graph $G$, let $V(G)$ and $E(G)$ denote the vertex set and the edge set of $G$ respectively. The Boolean function graph $B(G, L(G), \mathop {\mathrm NINC})$ of $G$ is a graph with vertex set $V(G)\cup E(G)$ and two vertices in $B(G, L(G), \mathop {\mathrm NINC})$ are adjacent if and only if they correspond to two adjacent vertices of $G$, two adjacent edges of $G$ or to a vertex and an edge not incident to it in $G$. For brevity, this graph is denoted by $B_{1}(G)$. In this paper, we determine domination number, independent, connected, total, cycle, point-set, restrained, split and non-split domination numbers of $B_{1}(G)$ and obtain bounds for the above numbers.
[1] J. Akiyama, T. Hamada, I. Yoshimura: On characterizations of the middle graphs. TRU. Math. 11 (1975), 35–39. MR 0414436
[2] R. B. Allan, R. Laskar: On domination and independent domination numbers of a graph. Discrete Math. 23 (1978), 73–76. DOI 10.1016/0012-365X(78)90105-X | MR 0523402
[3] M. Behzad: A criterion for the planarity of the total graph of a graph. Proc. Camb. Philos. Soc. 63 (1967), 679–681. DOI 10.1017/S0305004100041657 | MR 0211896 | Zbl 0158.20703
[4] S. B. Chikkodimath, E. Sampathkumar: Semi total graphs II. Graph Theory Research Report, Karnatak University 2 (1973), 5–9.
[5] E. J. Cockayne, R. M. Dawes, S. T. Hedetniemi: Total domination in graphs. Networks 10 (1980), 211–219. DOI 10.1002/net.3230100304 | MR 0584887
[6] E. J. Cockayne, B. L. Hartnell, S. T. Hedetniemi, R. Laskar: Perfect domination in graphs. J. Comb. Inf. Syst. Sci. 18 (1993), 136–148. MR 1317698
[7] G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. Laskar, L. R. Markus: Restrained domination in graph. Discrete Math. 203 (1999), 61–69. DOI 10.1016/S0012-365X(99)00016-3 | MR 1696234
[8] T. Hamada, I. Yoshimura: Traversability and connectivity of the middle graph of a graph. Discrete Math. 14 (1976), 247–256. DOI 10.1016/0012-365X(76)90037-6 | MR 0414435
[9] F. Harary: Graph Theory. Addison-Wesley, Reading, Mass., 1969. MR 0256911 | Zbl 0196.27202
[10] T. N. Janakiraman, S. Muthammai, M. Bhanumathi: On the Boolean function graph of a graph and on its complement. Math. Bohem. 130 (2005), 113–134. MR 2148646
[11] V. R. Kulli, B. Janakiram: The total global domination number of a graph. Indian J. Pure Appl. Math. 27 (1996), 537–542. MR 1390876
[12] O. Ore: Theory of Graphs. Amer. Math. Soc. Colloq. Publ. 38, Providence, AMS, 1962.
[13] E. Sampathkumar, L. Pushpa Latha: Point-set domination number of a graph. Indian J. Pure Appl. Math. 24 (1993), 225–229. MR 1218532
[14] E. Sampathkumar, H. B. Walikar: The connected domination number of a graph. J. Math. Phys. Sci. 13 (1979), 607–613. MR 0575817
[15] D. V. S. Sastry, B. Syam Prasad Raju: Graph equations for line graphs, total graphs, middle graphs and quasi-total graphs. Discrete Math. 48 (1984), 113–119. DOI 10.1016/0012-365X(84)90137-7 | MR 0732207
[16] H. Whitney: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54 (1932), 150–168. DOI 10.2307/2371086 | MR 1506881 | Zbl 0003.32804
Partner of
EuDML logo