2016/02/07 by Martin Milanič, Irena Penev, Milanič, Martin +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1602.02406
arxiv created 2016/02/07 · arxiv updated 2016/02/09
An ISK4 in a graph G is an induced subgraph of G that is isomorphic to a subdivision of K4 (the complete graph on four vertices). A wheel is a graph that consists of a chordless cycle, together with a vertex that has at least three neighbors in the cycle. A graph is ISK4,wheel-free if it has no ISK4 and does not contain a wheel as an induced subgraph. A "trigraph" is a generalization of a graph in which some pairs of vertices have "undetermined" adjacency. We prove a decomposition theorem for ISK4,wheel-free trigraphs. Our proof closely follows the proof of a decomposition theorem for ISK4-free graphs due to Lévêque, Maffray, and Trotignon (On graphs with no induced subdivision of K4. J. Combin. Theory Ser. B, 102(4):924-947, 2012).