Excluding a bipartite circle graph from line graphs

Abstract

We prove that for fixed bipartite circle graph H, all line graphs with sufficiently large rank-width (or clique-width) must contain an isomorphic copy of H as a pivotminor. To prove this, we introduce graphic delta-matroids. Graphic delta-matroids are minors of delta-matroids of line graphs and they generalize graphic or cographic matroids. IntroductionRobertson and Seymour [20] proved that every graph of sufficiently large tree-width must contain a minor isomorphic to a fixed planar graph. Their theorem was generalized to a theorem on representable matroids by Geelen, Gerards, and Whittle [9], stating that every matroid representable over a fixed finite field of sufficiently large branch-width must contain a minor isomorphic to a fixed planar matroid. (A planar matroid is a cycle matroid of a planar graph.)We aim to prove the following conjecture, that is another generalization of Robertson and Seymour's grid theorem. Rank-width is a graph width parameter, like tree-width, introduced by Oum and Seymour [18] to investigate clique-width [5]. Pivot-minors of a graph G are graphs obtained from G by repeatedly applying certain operations, like minors [17]. We defer definitions of rank-width and pivot-minor to Section 2. A circle graph is the intersection graph of chords in a circle, see Figure 1. Bipartite circle graphs are related to planar graphs, shown by de Fraysseix [6]. (See Lemma 23.) * sangil@math.gatech.edu † Partially supported by NSF grant 0354742

Rank‐width is less than or equal to branch‐width

“…We remark that with different methods, Oum [7] showed that the rank-width of L(G) is exactly one of bw(G) − 2, bw(G) − 1, or bw(G) if G is 2-connected.…”

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

“…One may wish to have a structure theorem describing graphs with no fixed vertex-minors or no fixed pivot-minors in order to extend these theorems to other forbidden graphs. Indeed, Oum [28] conjectured the following. A graph is a circle graph if it is an intersection graph of chords in a circle.…”

“…Vertex-minors and pivot-minors are graph containment relations introduced by Bouchet [3,4,5,6] while conducting research of circle graphs (intersection graphs of chords in a cycle) and 4-regular Eulerian digraphs. Furthermore, these graph operations have been used for developing theory on rank-width [20,26,27,28,29]. We review these concepts in Section 2.…”

Circle graphs and monadic second-order logic

“…(For rank-width and vertex-minors, see Oum [33].) This conjecture is proved for line graphs by Oum [34].…”