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).
We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.
We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an…
We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid $E_4$ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of $P_9$, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geome…
A plane near-triangulation G can be decomposed into a collection of induced subgraphs, described here as the W-components of G, such that G is perfect (respectively, chordal) if and only if each of its W-components is perfect (respectively, chordal). Each W-component is a 2-connected plane near-triangulation, free of edge separators and separating triangles. Graphs satisfying these conditions will be called W-near-triangulations. A linear time decomposition of G into its W-components is achievable using known techniques from the literature. W-near-triangulations have the property that the open…
We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vsković. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-h…
A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots b_1$, $P_2 = a \dots b_2$, $P_3 = a \dots b_3$ of length at least~1, two of which have length at least 2, vertex-disjoint except at $a$, and such that $b_1b_2b_3$ is a triangle and no edges exist between the paths except those of the triangle and the three edges incident with $a$. We give a polynomial time algorithm to compute a maximum weighted independent se…
