Generalized line graphs: Cartesian products and complexity of recognition

Aparna Lakshmanan S.¹ ; Csilla Bujtás² ; Zsolt Tuza³

¹St. Xavier's College for Women Aluva, Kerala
²University of Pannonia Veszprém
³Hungarian Academy of Sciences Budapest

The electronic journal of combinatorics, Tome 22 (2015) no. 3

Résumé

Putting the concept of line graph in a more general setting, for a positive integer $k$, the $k$-line graph $L_k(G)$ of a graph $G$ has the $K_k$-subgraphs of $G$ as its vertices, and two vertices of $L_k(G)$ are adjacent if the corresponding copies of $K_k$ in $G$ share $k-1$ vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The $k$-anti-Gallai graph $\triangle_k(G)$ of $G$ is a specified subgraph of $L_k(G)$ in which two vertices are adjacent if the corresponding two $K_k$-subgraphs are contained in a common $K_{k+1}$-subgraph in $G$.We give a unified characterization for nontrivial connected graphs $G$ and $F$ such that the Cartesian product $G\Box F$ is a $k$-line graph. In particular for $k=3$, this answers the question of Bagga (2004), yielding the necessary and sufficient condition that $G$ is the line graph of a triangle-free graph and $F$ is a complete graph (or vice versa). We show that for any $k\ge 3$, the $k$-line graph of a connected graph $G$ is isomorphic to the line graph of $G$ if and only if $G=K_{k+2}$. Furthermore, we prove that the recognition problem of $k$-line graphs and that of $k$-anti-Gallai graphs are NP-complete for each $k\ge 3$.

Détail

DOI : 10.37236/3983

Classification : 05C76, 05C62, 68Q17
Mots-clés : triangle graph, \(k\)-line graph, anti-Gallai graph, Cartesian product graph