Geodesic
On the Boolean function graph of a graph and on its complement
Mathematica Bohemica, Tome 130 (2005) no. 2, pp. 113-134

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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, structural properties of $B_1(G)$ and its complement including traversability and eccentricity properties are studied. In addition, solutions for Boolean function graphs that are total graphs, quasi-total graphs and middle graphs are obtained.
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, structural properties of $B_1(G)$ and its complement including traversability and eccentricity properties are studied. In addition, solutions for Boolean function graphs that are total graphs, quasi-total graphs and middle graphs are obtained.
DOI : 10.21136/MB.2005.134130
Classification : 05C12, 05C15, 05C45, 05C75, 06E30
Keywords: eccentricity; self-centered graph; middle graph; Boolean function graph 
Janakiraman, T. N.; Muthammai, S.; Bhanumathi, M. On the Boolean function graph of a graph and on its complement. Mathematica Bohemica, Tome 130 (2005) no. 2, pp. 113-134. doi: 10.21136/MB.2005.134130
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[1] J. Akiyama, T. Hamada, I. Yoshimura: On characterizations of the middle graphs. TRU Math. 11 (1975), 35–39. | MR

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

[3] J. A. Bondy, U. S. Murty: Graph Theory with Application. Macmillan, London, 1976.

[4] S. B. Chikkodimath, E. Sampathkumar: Semi-total graphs II. Graph Theory Research Report, Karnatak University 2 (1973), 5–9.

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

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

[7] E. Sampathkumar, Prabha S. Neeralagi: The neighborhood number of a graph. Indian J. Pure Appl. Math. 16 (1985), 126–132. | MR

[8] 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

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

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