Geodesic
Domination numbers on the complement of the Boolean function graph of a graph
Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 247-263

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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, point-set, restrained, split and non-split domination numbers in the complement $\bar{B}_{1}(G)$ of $B_{1}(G)$ and obtain bounds for the above numbers.
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, point-set, restrained, split and non-split domination numbers in the complement $\bar{B}_{1}(G)$ of $B_{1}(G)$ and obtain bounds for the above numbers.
DOI : 10.21136/MB.2005.134098
Classification : 05C15, 05C69
Keywords: domination number; eccentricity; radius; diameter; neighborhood; perfect matching; Boolean function graph 
Janakiraman, T. N.; Muthammai, S.; Bhanumathi, M. Domination numbers on the complement of the Boolean function graph of a graph. Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 247-263. doi: 10.21136/MB.2005.134098
@article{10_21136_MB_2005_134098,
     author = {Janakiraman, T. N. and Muthammai, S. and Bhanumathi, M.},
     title = {Domination numbers on the complement of the {Boolean} function graph of a graph},
     journal = {Mathematica Bohemica},
     pages = {247--263},
     year = {2005},
     volume = {130},
     number = {3},
     doi = {10.21136/MB.2005.134098},
     mrnumber = {2164655},
     zbl = {1111.05076},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134098/}
}
TY  - JOUR
AU  - Janakiraman, T. N.
AU  - Muthammai, S.
AU  - Bhanumathi, M.
TI  - Domination numbers on the complement of the Boolean function graph of a graph
JO  - Mathematica Bohemica
PY  - 2005
SP  - 247
EP  - 263
VL  - 130
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134098/
DO  - 10.21136/MB.2005.134098
LA  - en
ID  - 10_21136_MB_2005_134098
ER  - 
%0 Journal Article
%A Janakiraman, T. N.
%A Muthammai, S.
%A Bhanumathi, M.
%T Domination numbers on the complement of the Boolean function graph of a graph
%J Mathematica Bohemica
%D 2005
%P 247-263
%V 130
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134098/
%R 10.21136/MB.2005.134098
%G en
%F 10_21136_MB_2005_134098

[1] J. Akiyama, T. Hamada, I. Yoshimura: On characterizations of the middle graphs. Tru. Math. (1975), 35–39. | MR

[2] R. B. Allan, R. Laskar: On domination and independent domination of a graph. Discrete Math. 23 (1978), 73–76. | DOI | MR

[3] M. Behzad: A criterion for the planarity of the total graph of a graph. Proc. Cambridge Philos. Soc. 63 (1967), 679–681. | MR | Zbl

[4] S. B. Chikkodimath, E. Sampathkumar: Semi total graphs-II. Graph Theory Research Report, Karnatak University (1973), 5–9.

[5] E. J. Cockayne, R. M. Dawes, S. T. Hedetniemi: Total domination in graphs. Networks 10 (1980), 211–219. | DOI | MR

[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

[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 | MR

[8] T. Hamada, I. Yoshimura: Traversability and connectivity of the middle graph of a graph. Discrete Math. 14 (1976), 247–256. | DOI | MR

[9] F. Harary: Graph Theory. Addison-Wesley, Reading, Mass., 1969. | MR | Zbl

[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

[11] V. R. Kulli, B. Janakiram: The non-split domination number of a graph. Indian J. Pure Appl. Math. 27 (1996), 537–542. | MR

[12] O. Ore: Theory of Graphs. Amer. Math. Soc. Colloq. Publ., 38, Providence, 1962. | Zbl

[13] E. Sampathkumar, L. Pushpalatha: Point-set domination number of a graph. Indian J. Pure Appl. Math. 24 (1993), 225–229. | MR

[14] E. Sampathkumar, H. B. Walikar: The connected domination number of a graph. J. Math. Phys. Sci. 13, 607–613. | MR

[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 | MR

[16] H. Whitney: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54 (1932), 150–168. | DOI | MR | Zbl

Cité par Sources :