Vertex-minors of graphs Sang-il Oum (엄상일) IBS Discrete Mathematics Group / KAIST International Conference on Network Games, Tropical Geometry, and Quantum Communication Berlin June 6, 2019 Quantum Information Theory and Graph Theory Graph States |G> Graphs 3 operations to reduce some meaningful operations Vertex-minors (local Clifford operations, Local complementation Pauli Z measurements on vertices) deletion Schmit-rank width rank-width Plan • Planar graphs and minors • Circle graphs and vertex-minors • Connectivity Functions • Width Parameters • Extending the graph minors project of Robertson and Seymour • Weakening of conjectures on induced subgraphs Planar graphs and minors A graph G is planar if it can be drawn on the plane with no edge crossing. Operations that preserve planarity edge deletion G\e H is a minor of G if v e H is obtained from G by a sequence of deleting edges/vertices and contracting edges contraction G/e Easy Observation: vertex deletion If H is a minor of a planar graph G, then H is planar. G∖v Kuratowski's theorem on planar graphs (1930) Theorem (Kuratowski 1930 / Wagner 1937) G is planar if and only if G has no minor isomorphic to K5 or K3,3 K5 K3,3 Two minor-minimal non-planar graphs Circle graphs and vertex-minors / pivot-minors Circle graph = intersection graph of chords of a circle 1 2 1 5 chords vertices 3 3 2 3 intersect edges 4 1 4 2 5 4 5 A minor of a circle graph is not necessarily a circle graph. K6 is a circle graph Impossible to minor characterize circle graphs in terms of forbidden minors W5 is not a circle graph Local complementation and vertex-minors 2 1 2 1 1 6 5 1 6 6 6 5 3 3 3 3 2 3 2 3 1 4 1 4 6 6 4 2 5 4 2 5 4 5 4 5 G G * 1 G G * 3 "local complementation at 1" If G is a circle graph, then G*v is a circle graph. H is a vertex-minor of G if H can be Easy Observation: obtained from G by a sequence of If H is a vertex-minor of G and local complementations and G is a circle graph, vertex deletions then H is a circle graph. Pivot and pivot-minors 1 1 2 1 2 4 6 6 6 5 6 6 1 2 3 3 2 3 3 3 3 1 4 6 5 4 4 5 4 5 4 2 5 1 2 5 G G ∧ 23 If G is a circle graph, "pivot an edge 23" then G∧vw is a circle graph. H is a pivot-minor of G if H can be Easy Observation: obtained from G by a sequence of If H is a pivot-minor of G and pivots and G is a circle graph, vertex deletions then H is a circle graph. Every pivot-minor is a vertex-minor 6 2 1 1 2 4 2 2 4 5 6 6 6 6 6 5 1 1 3 3 3 3 3 3 3 3 1 4 1 4 4 5 5 4 6 64 1 4 2 5 2 5 2 5 1 2 5 G G * 3 G * 3 * 2 G * 3 * 2 * 3 = G ∧ 23 In general, G ∧ uv = G * u * v * u Every pivot-minor is a vertex-minor. (The converse is false.) CIRCLE GRAPH OBSTRUCTIONS UNDER PIVOTING JIM GEELEN AND SANG-IL OUM Abstract. A circle graph is the intersection graph of a set of chords of a circle. The class of circle graphs is closed under pivot- minors. We determine the pivot-minor-minimal non-circle-graphs; there are 15 obstructions. These obstructions are found, by com- puter search, as a corollary to Bouchet’s characterization of circle graphs under local complementation. Our characterization gener- alizes Kuratowski’s Theorem. 1. Introduction The class of circle graphs is closed with respect to vertex-minors and hence also pivot-minors. (Definitions are postponed until Section 2.) Bouchet [5]Forbidden gave the vertex-minors following characterization for circle graphs of circle graphs; the graphs W5, W7, and BW3 are defined in Figure 1. All graphs in this paper areTheorem simple; graphs(Bouchet have 1994) no loops and no parallel edges. G is a circle graph Theorem if 1.1and only(Bouchet) if . A graph is a circle graph if and only it has no vertex-minor G has no that vertex-minor is isomorphic isomorphic to W to5, W7, or BW3. W5 W7 BW3 Figure 1. W , W , and BW : Excluded vertex-minors Three5 vertex-minor7 minimal3 non-circle graphs for circle graphs. As a corollary to Bouchet’s theorem we prove the following result. Date: April 11, 2007 submitted; August 2008 revised. 1991 Mathematics Subject Classification. 05B35. Key words and phrases. circle graphs, delta-matroids, pivoting, principally uni- modular matrices, eulerian delta-matroids. The first author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author was partially supported by the SRC Program of Korea Sci- ence and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01002-0). 1 Forbidden pivot-minors for circle graphs Theorem (Geelen, O. 2009) G is a circle graph if and only if G has no pivot-minor isomorphicCIRCLE GRAPH OBSTRUCTIONS to 3 FIGURE 2. Excluded pivot-minors for circle graphs. FIGURE 3. H1, H2, and Q3. fundamental graphs of regular matroids. Seymour’s decomposition Theorem [11] provides a good characterization and a recognition algorithm for regular matroids and we hope to obtain similar results for PU-orientable graphs. Bouchet [2] proved that circle graphs admit PU-orientations and we hope that the class of circle graphs will play a central role in a decomposition theorem for PU-orientable graphs. The class of PU-orientable graphs is closed under pivot-minors but not under vertex-minors, and hence it is desirable to have the excluded pivot-minors for the class of circle graphs. Although the class of PU-orientable graphs is not closed under local comple- mentation, Bouchet’s theorem does imply the following curious connection between PU-orientability and circle graphs: a graph is a circle graph if and only if every locally equivalent graph is PU-orientable. We prove Theorem 1.2 by studying the graphs that are pivot-minor-minimal while containing a vertex-minor isomorphic to one of W5, W7, or BW3. We require the following two lemmas that are proved in Section 3. The proofs are direct but inelegant. These facts are transparent in the context of isotropic systems; see Bouchet [3]. However, the direct proofs are shorter than the requisite introduction to isotropic systems. Journal of Graph Theory DOI 10.1002/jgt 34 H. de Fmyssek Fig. 2. Let C be an isomorphic inrape of the: unit circle S1 drawn close to the base 8 such that C is the border of a ubuiar neighbourhood of B (see Fig. 2). The chords corresponding to the edges in B’- are obtained by restricting these edges to the outside of the disc delimited by C. .,4 chord corresponding to 8n edge e in B &ins the two arcs of C drawn parallely along e. The chords intersecting e are obviously the ones corresponding to the funda- mental cocircuit associated with e. (2) Let G be a bipartite chordable graph. The two sets of chords, correspond- ing to the bipartition of the set of vertices of G, can be represented by two systems of chords inside and outside the unit circle S’ (see Fig. 3). The chords drawn inside S” partition the disc delimited by S’ into faces {I$}isr in the center of which we place a point Vi.The set {Ui}<,=Iis the set of vertices of the graphical representation PW of W(G) whose edges are drawn in the following 3W4i 3 yI H. de Fmyssek First join the vertices of PW corresponding to adjacent faces. Let B be the resulting set of edges. The set BL of other edges of PW is obtained by extending the outside chords to the vertices t+, Uj corresponding to the faces Fi, 4 to which they are incident. One can easily check that B is a tree of PW (resp. BA a cotree of PW), that is, a base of W(G) (resp. a cobase of W(G)) and that G is the fundamenta! interlacement graph associated with the base B of W(G), Fig. 2. Let C be an isomorphic inrape of the: unit circle S1 drawn close to the base 8 such that C is the border of a ubuiar neighbourhood of B (see Fig. 2). The chords corresponding to the edges in B’- are obtained by restricting these 2 edges to the outside of the disc delimited by C. .,4 chord corresponding to 8n edge Theoreme in (de B &insFraysseix the two 1981) arcs of C drawn parallely along e. The chords intersecting e are obviously the ones corresponding to the funda- A bipartite graph is a circle graph Fig. 3. mental cocircuit associated with e. if and only if it is a fundamental graph of a planar graph. (2) Let G be a bipartite chordable graph. The two sets of chords, correspond- ing to the bipartition of the set of vertices of G, can be represented by two systems of chords inside and outside the unit circle S’ (see Fig. 3). The chords drawn inside S” partition the disc delimited by S’ into faces {I$}isr in the center of which we place a point Vi.The set {Ui}<,=Iis the set of vertices of the graphical representation PW of W(G) whose edges are drawn in the following Wi3 yI First join the vertices of PW corresponding to adjacent faces.