Structure and Generation of Crossing-critical Graphs
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1 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Structure and Generation of Crossing-critical Graphs Petr Hliněný Faculty of Informatics, Masaryk University Brno, Czech Republic joint work with Zdeněk Dvořák and Bojan Mohar
2 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane?
3 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok
4 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok no ok
5 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok no ok What forces high crossing number?
6 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok no ok What forces high crossing number? Many edges cf. Euler s formula, and some strong enhancements [Ajtai, Chvátal, Newborn, Szemeredi, 1982; Leighton].
7 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok no ok What forces high crossing number? Many edges cf. Euler s formula, and some strong enhancements [Ajtai, Chvátal, Newborn, Szemeredi, 1982; Leighton]. Structural properties (even with sparse edges) but what exactly?
8 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 1 Crossing Number and Crossing-critical Crossing number cr(g) : how many edge crossings are required to draw G in the plane? no ok no ok What forces high crossing number? Many edges cf. Euler s formula, and some strong enhancements [Ajtai, Chvátal, Newborn, Szemeredi, 1982; Leighton]. Structural properties (even with sparse edges) but what exactly? Definition. Graph H is c-crossing-critical if cr(h) c and cr(h e) < c for all edges e E(H). We study crossing-critical graphs to understand what structural properties force the crossing number of a graph to be large.
9 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Some starting examples Kuratowski (30): The only 1-crossing-critical graphs K 5 and K 3,3. (Yes, up to subdivisions, but we ignore that... )
10 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Some starting examples Kuratowski (30): The only 1-crossing-critical graphs K 5 and K 3,3. (Yes, up to subdivisions, but we ignore that... ) Širáň (84), Kochol (87): Infinitely many c-crossing-critical graphs for every c 2, even simple 3-connected.
11 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs And some more recent constructions Salazar (03):
12 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs And some more recent constructions Salazar (03): Hliněný (02):
13 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs... and a bit of surprise Dvořák, Mohar (10): A c-crossing-crit. graph with unbounded degree, c 171.
14 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c + 16.
15 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c Geelen, Richter, Salazar (04): A c-crossing-critical graph has tree-width bounded in c.
16 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c Geelen, Richter, Salazar (04): A c-crossing-critical graph has tree-width bounded in c. Hliněný (03):... and also path-width bounded in c.
17 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c Geelen, Richter, Salazar (04): A c-crossing-critical graph has tree-width bounded in c. Hliněný (03):... and also path-width bounded in c. Hliněný and Salazar (08): A c-crossing-critical graph has no large K 2,n -subdivision.
18 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c Geelen, Richter, Salazar (04): A c-crossing-critical graph has tree-width bounded in c. Hliněný (03):... and also path-width bounded in c. Hliněný and Salazar (08): A c-crossing-critical graph has no large K 2,n -subdivision. Bokal, Oporowski, Richter, Salazar (16): Fully described 2-crossing-critical graphs up to fin. small exceptions.
19 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 2 Properties of Crossing-Critical Graphs Richter and Thomassen (93): A c-crossing-critical graph has cr(g) 2.5c Geelen, Richter, Salazar (04): A c-crossing-critical graph has tree-width bounded in c. Hliněný (03):... and also path-width bounded in c. Hliněný and Salazar (08): A c-crossing-critical graph has no large K 2,n -subdivision. Bokal, Oporowski, Richter, Salazar (16): Fully described 2-crossing-critical graphs up to fin. small exceptions. Dvořák, Hliněný, Mohar, Postle (11, not published): A c-crossing-critical graph cannot contain a deep nest, and so it has bounded dual diameter.
20 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs What kinds of crossing-critical graphs do we have? Informally, thin and long bands, joined together, and huge faces around... + combinations of these together
21 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs What kinds of crossing-critical graphs do we have? Informally, thin and long bands, joined together, and huge faces around... * Our result * + combinations of these together I. Nothing else than the previous can constitute crossing-criticality.
22 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs What kinds of crossing-critical graphs do we have? Informally, thin and long bands, joined together, and huge faces around... * Our result * + combinations of these together I. Nothing else than the previous can constitute crossing-criticality. II. There are well-defined local operations (replacements) that can reduce any large c-crossing-critical graph to a smaller one.
23 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs What kinds of crossing-critical graphs do we have? Informally, thin and long bands, joined together, and huge faces around... * Our result * + combinations of these together I. Nothing else than the previous can constitute crossing-criticality. II. There are well-defined local operations (replacements) that can reduce any large c-crossing-critical graph to a smaller one. III. There are finitely many well-defined building bricks that can produce all c-crossing-critical graphs from a finite set of base graphs.
24 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Once again, with an informal explanation I. Nothing else than these can constitute crossing-criticality for sufficiently large graphs.
25 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Once again, with an informal explanation I. Nothing else than these can constitute crossing-criticality for sufficiently large graphs. II. There are well-defined local operations (replacements) that can reduce any large c-crossing-critical graph to a smaller one.
26 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs Once again, with an informal explanation I. Nothing else than these can constitute crossing-criticality for sufficiently large graphs. II. There are well-defined local operations (replacements) that can reduce any large c-crossing-critical graph to a smaller one. III. There are finitely many well-defined building bricks that can produce all c-crossing-critical graphs from a finite set of base graphs.
27 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either
28 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either find a spec. substructure, not relevant to crossing-crit. graphs,
29 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either find a spec. substructure, not relevant to crossing-crit. graphs, or, get a topological long-band structure composed of boundedsize tiles separated (between consecutive ones) by paths.
30 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either find a spec. substructure, not relevant to crossing-crit. graphs, or, get a topological long-band structure composed of boundedsize tiles separated (between consecutive ones) by paths. 2. Removing and inserting tiles in a plane band: Get a long plane band in our crossing-crit. graph, as in the previous.
31 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either find a spec. substructure, not relevant to crossing-crit. graphs, or, get a topological long-band structure composed of boundedsize tiles separated (between consecutive ones) by paths. 2. Removing and inserting tiles in a plane band: Get a long plane band in our crossing-crit. graph, as in the previous. Find repeated isomorphic sections, and shorten the band between suitable two consecutive repetitions.
32 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3 Towards the Structural Description Dividing the proof into two major steps. 1. General understanding of the struct. of a plane band and tiles: In every plane (topological!) graph of bounded path-width, either find a spec. substructure, not relevant to crossing-crit. graphs, or, get a topological long-band structure composed of boundedsize tiles separated (between consecutive ones) by paths. 2. Removing and inserting tiles in a plane band: Get a long plane band in our crossing-crit. graph, as in the previous. Find repeated isomorphic sections, and shorten the band between suitable two consecutive repetitions. Prove that such shortening preserves crossing-criticality.
33 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture).
34 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture). a) Modify the decompos. to ensure homogeneous horizon. connectivity.
35 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture). a) Modify the decompos. to ensure homogeneous horizon. connectivity. b) Characterize a bounded topological type of each bag.
36 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture). a) Modify the decompos. to ensure homogeneous horizon. connectivity. b) Characterize a bounded topological type of each bag. Apply an algebraic tool Simon s factorization forest, to a semigroup formed by concatenation of these topological types.
37 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture). a) Modify the decompos. to ensure homogeneous horizon. connectivity. b) Characterize a bounded topological type of each bag. Apply an algebraic tool Simon s factorization forest, to a semigroup formed by concatenation of these topological types. c) The previous gives a subband with a homogeneous topol. structure ; either the desired band with properly separated and connected tiles, or
38 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.1 Structure of a plane band and its tiles Starting from a path-decomposition of bounded width, the main trouble is that its bags do not correspond to our topological graph (our picture). a) Modify the decompos. to ensure homogeneous horizon. connectivity. b) Characterize a bounded topological type of each bag. Apply an algebraic tool Simon s factorization forest, to a semigroup formed by concatenation of these topological types. c) The previous gives a subband with a homogeneous topol. structure ; either the desired band with properly separated and connected tiles, or one of special substructures forbidden in crossing-critical graphs: face
39 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.2 Removing and inserting tiles a) Long band consider shelled bands, shelled fans, and necklaces.
40 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.2 Removing and inserting tiles a) Long band consider shelled bands, shelled fans, and necklaces. b) Repeated isomorphic sections overlay, and forget the stretch betw. G = 2... G 1
41 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.2 Removing and inserting tiles a) Long band consider shelled bands, shelled fans, and necklaces. b) Repeated isomorphic sections overlay, and forget the stretch betw. G = 2... G 1 c) Use further repetitions of this local picture around to argue that c-crossing-criticality is preserved:
42 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 3.2 Removing and inserting tiles a) Long band consider shelled bands, shelled fans, and necklaces. b) Repeated isomorphic sections overlay, and forget the stretch betw. G = 2... G 1 c) Use further repetitions of this local picture around to argue that c-crossing-criticality is preserved: G 1 drawn with < c crossings can expand with no new crossing, (more difficult) G e drawn with < c crossings can modify and shrink to G 1 e with no new crossing.
43 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)?
44 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs.
45 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs. Unfortunately, very unlikely also for our building bricks, since the crossing number of a twisted planar tile is NP-hard.
46 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs. Unfortunately, very unlikely also for our building bricks, since the crossing number of a twisted planar tile is NP-hard. What further applications of our characterization can we have?
47 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs. Unfortunately, very unlikely also for our building bricks, since the crossing number of a twisted planar tile is NP-hard. What further applications of our characterization can we have? A new view of known properties, such as the following one: the average degree of an infinite c-crossing-critical family is bounded away from 3 below and 6 above.
48 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs. Unfortunately, very unlikely also for our building bricks, since the crossing number of a twisted planar tile is NP-hard. What further applications of our characterization can we have? A new view of known properties, such as the following one: the average degree of an infinite c-crossing-critical family is bounded away from 3 below and 6 above. And some currently open problems, such as that the crossing number of a c-crossing-critical graph should be c+o( c), and whether there exists a 5-regular c-crossing-critical family.
49 Petr Hliněný, SoCG, Budapest, / 12 Structure of crossing-critical graphs 4 Final Remarks Could our result be as nice as the one for 2-crossing-critical? That is, will our characterization eventually be explicit (wrt. c)? Expectedly, not for the small base graphs. Unfortunately, very unlikely also for our building bricks, since the crossing number of a twisted planar tile is NP-hard. What further applications of our characterization can we have? A new view of known properties, such as the following one: the average degree of an infinite c-crossing-critical family is bounded away from 3 below and 6 above. And some currently open problems, such as that the crossing number of a c-crossing-critical graph should be c+o( c), and whether there exists a 5-regular c-crossing-critical family. Thank you for your attention.
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