Generalized line graphs: Cartesian products and complexity of recognition

Putting the concept of line graph in a more general setting, for a positive integer k the k-line graph Lk(G) of a graph G has the Kk-subgraphs of G as its vertices, and two vertices of Lk(G) are adjacent if the corresponding copies of Kk 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 Δk(G) of G is a specified subgraph of Lk(G) in which two vertices are adjacent if the corresponding two Kk-subgraphs are contained in a common Kk+1-subgraph in G. We give a unified characterization for nontrivial connected graphs G and F such that the Cartesian product G□F is a k-line graph. In particular for k = 3, this answers the question of Bagga (2004), yielding the necessary and suficient 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 ≥ 3, the k-line graph of a connected graph G is isomorphic to the line graph of G if and only if G = Kk+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 ≥ 3. © 2015, Australian National University. All rights reserved

Induced cycles in triangle graphs

The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of $C_n$-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs $G$ for which ${\cal T}(G)$ is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.Comment: 27 page