Graphs with no 7-wheel subdivision Rebecca Robinson 1
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1 Graphs with no 7-wheel subdivision Rebecca Robinson Monash University (Clayton Campus) (joint work with Graham Farr) Graphs with no 7-wheel subdivision Rebecca Robinson 1
2 TOPOLOGICAL CONTAINMENT 1 Topological containment G Y X topologically contains contains an -subdivision Graphs with no 7-wheel subdivision Rebecca Robinson 2
3 , does TOPOLOGICAL CONTAINMENT Applications of topological containment Forest does not contain any -subdivisions Planar graph does not contain any (Kuratowski, 1930) -subdivisions or -subdivisions Series-parallel graph does not contain any Problem of topological containment: -subdivisions (Duffin, 1965) For some fixed pattern graph -subdivision? : given a graph contain an Graphs with no 7-wheel subdivision Rebecca Robinson 3
4 such that ROBERTSON AND SEYMOUR RESULTS 2 Robertson and Seymour results DISJOINT PATHS (DP) Input: Graph ; pairs of vertices of. Question: Do there exist paths of, mutually vertex-disjoint, joins and? Graphs with no 7-wheel subdivision Rebecca Robinson 4
5 ROBERTSON AND SEYMOUR RESULTS DISJOINT PATHS is in P for any fixed. This implies topological containment problem for fixed repeatedly. is also in P use DP We know p-time algorithms must exist for topological containment, but practical algorithms not given huge constants. Graphs with no 7-wheel subdivision Rebecca Robinson 5
6 PREVIOUS RESULTS 3 Previous results Theorem (Farr, 88). Let -subdivision if and only ifhas a vertex size at least 5 which does not contain. be 3-connected, with no internal 3-edge-cutset. Then has a of degree at least 5 and a circuit of : wheel with five spokes Graphs with no 7-wheel subdivision Rebecca Robinson 6
7 PREVIOUS RESULTS 2 2 Internal 3-edge-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 7
8 PREVIOUS RESULTS Theorem (Robinson & Farr, 2008). Let be a 3-connected graph that is not topologically contained in the graph. Suppose has no internal 3-edge-cutsets, no internal 4-edge-cutsets, and is a graph on which neither Reduction 1A nor Reduction 2A can be performed, for least 6. Then. has a -subdivision if and only ifhas a vertex of degree at Graphs with no 7-wheel subdivision Rebecca Robinson 8
9 PREVIOUS RESULTS Graph Graphs with no 7-wheel subdivision Rebecca Robinson 9
10 PREVIOUS RESULTS 3 3 Internal 4-edge-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 10
11 PREVIOUS RESULTS Definition If of vertex in is a subset of graph, then such that any two vertices of joined by a path in are. Each element of is referred to as a bridge of with no internal. denotes the set of all maximal subsets U 1 U 3 U 2 W G,, are all bridges of. Graphs with no 7-wheel subdivision Rebecca Robinson 11
12 PREVIOUS RESULTS u u P u Z P u Z Y v Y v P w P w X x w S w S Reduction 1A Graphs with no 7-wheel subdivision Rebecca Robinson 12
13 PREVIOUS RESULTS u u Z Z Y v Y v P w X x P w w w S S Reduction 2A Graphs with no 7-wheel subdivision Rebecca Robinson 13
14 PREVIOUS RESULTS Structure of graphs with no -subdivisions Graphs with no 7-wheel subdivision Rebecca Robinson 14
15 NEW RESULTS BASED ON THEOREM 4 New results based on theorem Theorem. Let a 3-connected graph with at least 12 vertices. Suppose be has no internal 3-edge-cutsets, no internal 4-edge-cutsets, and is a graph on which neither Reduction 1A nor Reduction 2A can be performed, for. Then has a -subdivision if and only if contains some vertex of degree at least 6. Graphs with no 7-wheel subdivision Rebecca Robinson 15
16 NEW RESULTS BASED ON THEOREM Theorem. Let a 3-connected graph with at least 14 vertices. Suppose be has no type 1, 2, 3, or 4 edge-vertex-cutsets, and is a graph on which Reductions 1A, 1B, 2A, and 2B cannot be performed, for. Let be a vertex of degree in. Then, either has a -subdivision centred on, or has a -subdivision centred on some vertex of degree. Graphs with no 7-wheel subdivision Rebecca Robinson 16
17 NEW RESULTS BASED ON THEOREM v >= 4 e 1 >= 4 e 2 G 1 G 2 Type 1 edge-vertex-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 17
18 NEW RESULTS BASED ON THEOREM >= 4 G 1 e G 2 >= 4 Type 2 edge-vertex-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 18
19 NEW RESULTS BASED ON THEOREM v >= 4 e 1 e 2 >= 4 e 3 e 4 G 1 G 2 Type 3 edge-vertex-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 19
20 NEW RESULTS BASED ON THEOREM >= 4 G 1 e 1 e 2 G 2 >= 4 Type 4 edge-vertex-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 20
21 NEW RESULT ON 5 New result on graphs Characterization (up to bounded size pieces) of graphs that do not contain -subdivisions Uses similar techniques to the and results Graphs with no 7-wheel subdivision Rebecca Robinson 21
22 be a 3-connected graph with at least 38 vertices. Suppose NEW RESULT ON Theorem. Let 3 or 4-edge-cutsets, no internal -cutsets... has no internal 5 5 Internal -cutset Graphs with no 7-wheel subdivision Rebecca Robinson 22
23 NEW RESULT ON Theorem. Let a 3-connected graph with at least 38 vertices. Suppose be has no internal 3 or 4-edge-cutsets, no internal 4a edge-vertex-cutsets... -cutsets, no type 1, 1a, 2, 2a, 3, 3a, 4, or v (deg < 7) >= 3 e 1 >= 3 e 2 G 1 G 2 Type 1a edge-vertex-cutset Graphs with no 7-wheel subdivision Rebecca Robinson 23
24 NEW RESULT ON Theorem. Let a 3-connected graph with at least 38 vertices. Suppose be has no internal 3 or 4-edge-cutsets, no internal -cutsets, no type 1, 1a, 2, 2a, 3, 3a, 4, or 4a edge-vertex-cutsets, and is a graph on which Reductions 1A, 1B, 1C, 2A, 2B, 3, 4, 5, and 6 cannot be performed, for... t t u u G X G X v X v w w S S Reduction 3 Graphs with no 7-wheel subdivision Rebecca Robinson 24
25 NEW RESULT ON W t Y W t Y P t u P t u P u P u v v P w P w X w Z w Z S S Reduction 4 Graphs with no 7-wheel subdivision Rebecca Robinson 25
26 NEW RESULT ON Y t Y t u u P u P u v X v Z w Z w S S Reduction 5 Graphs with no 7-wheel subdivision Rebecca Robinson 26
27 NEW RESULT ON Z t Z t u u P u1 P u1 v P u2 Y v P u2 Y X ( k) w S w S Reduction 6 Graphs with no 7-wheel subdivision Rebecca Robinson 27
28 NEW RESULT ON Theorem. Let a 3-connected graph with at least 38 vertices. Suppose be has no internal 3 or 4-edge-cutsets, no internal -cutsets, no type 1, 1a, 2, 2a, 3, 3a, 4, or 4a edge-vertex-cutsets, and is a graph on which Reductions 1A, 1B, 1C, 2A, 2B, 3, 4, 5, and 6 cannot be performed, for. Then has a -subdivision if and only if contains some vertex of degree at least 7. Graphs with no 7-wheel subdivision Rebecca Robinson 28
29 NEW RESULT ON Proof a summary. Suppose the conditions of the hypothesis hold for some graph. By the strengthened result, there exists some vertex of degree in that has a -subdivision centred on it. v 6 u How does connect to the rest of in order to preserve 3-connectivity? Graphs with no 7-wheel subdivision Rebecca Robinson 29
30 NEW RESULT ON Three possibilities: (a) Path from to some vertex on the rim of the -subdivision, not meeting any spoke. (b) Two paths from to two separate spokes of. (c) Path from to some vertex on one of the spokes of the such that this path that does not meet except at its end points. -subdivision, Graphs with no 7-wheel subdivision Rebecca Robinson 30
31 NEW RESULT ON Cases (a) and (c) are straightforward to deal with. Case (b) takes up most of the proof. All possible configurations in case (b) result in a three. -subdivision except for Graphs with no 7-wheel subdivision Rebecca Robinson 31
32 NEW RESULT ON = u 1 v 6 = u 2 Case (b)(i) Graphs with no 7-wheel subdivision Rebecca Robinson 32
33 NEW RESULT ON = u 1 v 6 = u 2 Case (b)(ii) Graphs with no 7-wheel subdivision Rebecca Robinson 33
34 NEW RESULT ON = u 1 v 6 u 2 Case (b)(iii) Graphs with no 7-wheel subdivision Rebecca Robinson 34
35 NEW RESULT ON These graphs meet the 3-connectivity requirements, but not the other requirements of the hypothesis: eg. forbidden reductions. So there must be more structure to the graphs. More in-depth case analysis required, based on different ways of adding this structure. C program to automate parts of this analysis; many parts of the proof depend on results generated by this program. The program: constructs the various simple graphs that arise as cases in the proof, and tests each graph for the presence of a -subdivision. Graphs with no 7-wheel subdivision Rebecca Robinson 35
36 NEW RESULT ON Case (b)(i): further detail 1. Path from to Four cases with no -subdivision v 6 v 6 Q Q v 6 v 6 Q Q Graphs with no 7-wheel subdivision Rebecca Robinson 36
37 from NEW RESULT ON 1.1. Path to 16 cases with no -subdivision v 6 R v 6 R Q Q v 6 v 6 Q R Q R Graphs with no 7-wheel subdivision Rebecca Robinson 37
38 NEW RESULT ON Path such that is not a separating set 64 cases with no -subdivision Q R 1 Q R 1 v 6 v 6 R R v3 Q v 6 Q R 1 v 6 R R 1 R Graphs with no 7-wheel subdivision Rebecca Robinson 38
39 NEW RESULT ON Path such that is not a separating set 256 cases with no -subdivision Q R 2 Q R 2 R 1 R 1 v 6 v 6 R R Q R 2 Q R 2 R 1 R 1 v6 R v 6 R Graphs with no 7-wheel subdivision Rebecca Robinson 39
40 NEW RESULT ON Path such that is not a separating set always results in a -subdivision v 6 R 3 Graphs with no 7-wheel subdivision Rebecca Robinson 40
41 NEW RESULT ON No such path: so forms a separating set. U 2 v 6 U 6 U 4 U u Graphs with no 7-wheel subdivision Rebecca Robinson 41
42 NEW RESULT ON It can be shown that either: a -subdivision exists centred on some vertex in U 2 v 6 U 6 U 4 U u Graphs with no 7-wheel subdivision Rebecca Robinson 42
43 NEW RESULT ON an internal -cutset exists in U 2 U 6 \ S 3 ( 5 vertices) U 4 U u Graphs with no 7-wheel subdivision Rebecca Robinson 43
44 NEW RESULT ON or Reduction 4 can be performed on U 2 v0 U 6 \ S 3 (< 5 vertices) U 2 U 4 U u U 4 U u Graphs with no 7-wheel subdivision Rebecca Robinson 44
45 NEW RESULT ON No path exists: forms a separating set. U 2 v6 U 4 S 2 Graphs with no 7-wheel subdivision Rebecca Robinson 45
46 NEW RESULT ON Suppose there are only two bridges of : and. Lemma. Let a 3-connected graph with at least 19 vertices. Suppose be has no internal 3 or 4-edge-cutsets, no type 1, 2, 2a, 3, 3a, or 4 edge-vertex-cutsets, and is a graph on which none of Reductions 1A, 1B, 1C, 2A, and 3 can be performed. Let be a separating set of vertices in such that is adjacent to both and, and such that there are exactly two bridges, and, of. Suppose that has at least four neighbours in. Then contains a -subdivision. These conditions hold, so must contain a -subdivision. Graphs with no 7-wheel subdivision Rebecca Robinson 46
47 NEW RESULT ON Suppose there exists a third bridge of. Then either: a -subdivision exists or one of the forbidden Reductions can be performed A U 2 U 2 U 4 v5 v 6 U 4 v5 v 6 Graphs with no 7-wheel subdivision Rebecca Robinson 47
48 NEW RESULT ON No path exists: forms a separating set. U 6 v 6 U2 S 1 Graphs with no 7-wheel subdivision Rebecca Robinson 48
49 NEW RESULT ON 1.2. No path exists: forms a separating set. W v 6 U 2 U(u) Graphs with no 7-wheel subdivision Rebecca Robinson 49
50 NEW RESULT ON 2. No path from to. v 6 U 2 U 4 U(u) Again, either a performed. W -subdivision exists, or a forbidden reduction can be Graphs with no 7-wheel subdivision Rebecca Robinson 50
51 NEW RESULT ON Structure of graphs with no -subdivisions First, must reduce a graph as much as possible, using the six forbidden reductions. The resulting graph in its reduced form must be composed of pieces that contain at least 38 vertices. Each piece must: be 3-connected; contain no internal 3- or 4-edge cutsets, or any of the other types of forbidden separating sets; and contain no vertices of degree. Graphs with no 7-wheel subdivision Rebecca Robinson 51
52 NEW RESULT ON Each of the pieces are joined together in a tree-like structure Each piece is joined to the rest of the graph so that either: there exists a separating set of size one piece from another; or, the removal of which disconnects there exists either an internal 3- or 4-edge cutset, or one of the forbidden separating sets, the removal of which disconnects one piece from another. Graphs with no 7-wheel subdivision Rebecca Robinson 52
53 NEW RESULT ON Graphs with no 7-wheel subdivision Rebecca Robinson 53
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