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

Homometric Number of Graphs

Given a graph G=(V,E), two subsets S_1 and S_2 of the vertex set V are homometric, if their distance multi sets are equal. The homometric number h(G) of a graph G is the largest integer k such that there exist two disjoint homometric subsets of cardinality k. We find lower bounds for the homometric number of the Mycielskian of a graph and the join and the lexicographic product of two graphs. We also obtain the homometric number of the double graph of a graph, the cartesian product of any graph with K_2 and the complete bipartite graph. We also introduce a new concept called weak homometric number and find weak homometric number of some graphs

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

Effects of antiplatelet therapy on stroke risk by brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases: subgroup analyses of the RESTART randomised, open-label trial

Background Findings from the RESTART trial suggest that starting antiplatelet therapy might reduce the risk of recurrent symptomatic intracerebral haemorrhage compared with avoiding antiplatelet therapy. Brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases (such as cerebral microbleeds) are associated with greater risks of recurrent intracerebral haemorrhage. We did subgroup analyses of the RESTART trial to explore whether these brain imaging features modify the effects of antiplatelet therapy

Clique Irreducibility of Some Iterative Classes of Graphs

In this paper, two notions, the clique irreducibility and clique vertex irreducibility are discussed. A graph G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and it is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G. It is proved that L(G) is clique irreducible if and only if every triangle in G has a vertex of degree two. The conditions for the iterations of line graph, the Gallai graphs, the anti-Gallai graphs and its iterations to be clique irreducible and clique vertex irreducible are also obtained