Faithful Representations of Graphs by Islands in the Extended Grid
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1 Faithful Representations of Graphs by Islands in the Extended Grid Michael D. Coury Pavol Hell Jan Kratochvíl Tomáš Vyskočil Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University
2 Grid The set of points in plane with integral coordinates.
3 Grid The set of points in plane with integral coordinates. Two distinct points A = (X A,Y A ) and B = (X B,Y B ) are adjacent, if X A = X B and Y A Y B 1 or Y A = Y B and X A X B 1.
4 Grid The set of points in plane with integral coordinates. Two distinct points A = (X A,Y A ) and B = (X B,Y B ) are adjacent, if X A = X B and Y A Y B 1 or Y A = Y B and X A X B 1.
5 Extended grid Like a grid but we add diagonal edges.
6 Extended grid Like a grid but we add diagonal edges.
7 Extended grid Like a grid but we add diagonal edges.
8 Island An island is a set of points in the extended grid which induces a connected subgraph.
9 Island An island is a set of points in the extended grid which induces a connected subgraph. connected, forms island not connected
10 Island adjacency Two islands i,j are adjacent if there is a pair of points P i, Q j, that are adjacent in the extended grid.
11 Island adjacency Two islands i,j are adjacent if there is a pair of points P i, Q j, that are adjacent in the extended grid.
12 Faithful representation A faithful representation of a graph G by islands is a set I of vertex disjoint islands in the extended grid, such that the adjacency graph of islands of I is isomorphic to G.
13 Faithful representation A faithful representation of a graph G by islands is a set I of vertex disjoint islands in the extended grid, such that the adjacency graph of islands of I is isomorphic to G.
14 Faithful representation A faithful representation of a graph G by islands is a set I of vertex disjoint islands in the extended grid, such that the adjacency graph of islands of I is isomorphic to G.
15 Motivation Motivation for this structure came from adiabatic quantum computation (AQC) where vertices of islands are qubits and edges are couplers.
16 Motivation Motivation for this structure came from adiabatic quantum computation (AQC) where vertices of islands are qubits and edges are couplers.
17 ISLANDS The set of graphs which have faithful representations by islands will be denoted ISLAND.
18 ISLANDS The set of graphs which have faithful representations by islands will be denoted ISLAND. An island with at most k vertices will be called a k-island.
19 ISLANDS The set of graphs which have faithful representations by islands will be denoted ISLAND. An island with at most k vertices will be called a k-island. The set of graphs which have faithful representations by k-islands will be denoted k-island
20 Connections ISLAND and k-island graphs are intersection graphs of connected regions in plane.
21 Connections ISLAND and k-island graphs are intersection graphs of connected regions in plane.
22 Connections ISLAND and k-island graphs are intersection graphs of connected regions in plane.
23 Induce subgraph of extended grid The graphs from 1-ISLAND are exactly induced subgraphs of extended grid.
24 Induce subgraph of extended grid The graphs from 1-ISLAND are exactly induced subgraphs of extended grid.
25 Induce subgraph of extended grid The graphs from 1-ISLAND are exactly induced subgraphs of extended grid.
26 String graphs STRING graphs are intersection graphs of curves in plane.
27 String graphs STRING graphs are intersection graphs of curves in plane.
28 String graphs STRING graphs are intersection graphs of curves in plane. STRING graphs have been introduced independently by Benzer(1959) and Sinden(1966).
29 String graphs STRING graphs are intersection graphs of curves in plane. STRING graphs have been introduced independently by Benzer(1959) and Sinden(1966). Ehrlich, Even and Tarjan (1976) showed computing the chromatic number of string graphs to be NP-hard.
30 String graphs STRING graphs are intersection graphs of curves in plane. STRING graphs have been introduced independently by Benzer(1959) and Sinden(1966). Ehrlich, Even and Tarjan (1976) showed computing the chromatic number of string graphs to be NP-hard. Kratochvíl (1991) showed string graph recognition to be NP-hard.
31 String graphs STRING graphs are intersection graphs of curves in plane. STRING graphs have been introduced independently by Benzer(1959) and Sinden(1966). Ehrlich, Even and Tarjan (1976) showed computing the chromatic number of string graphs to be NP-hard. Kratochvíl (1991) showed string graph recognition to be NP-hard. In that time it was not even known if this problem is decidable!!!
32 String graphs continued There are STRING graphs which need exponential number of crossings, it was proven by Kratochvíl and Matoušek (1991).
33 String graphs continued There are STRING graphs which need exponential number of crossings, it was proven by Kratochvíl and Matoušek (1991). The naive method to show that the recognition of STRING graphs is in NP is not working.
34 String graphs continued There are STRING graphs which need exponential number of crossings, it was proven by Kratochvíl and Matoušek (1991). The naive method to show that the recognition of STRING graphs is in NP is not working. Pach,Janos and Schaefer, Štefankovič independently proved in 2002 that recognition of STRING graphs is decidable.
35 String graphs continued There are STRING graphs which need exponential number of crossings, it was proven by Kratochvíl and Matoušek (1991). The naive method to show that the recognition of STRING graphs is in NP is not working. Pach,Janos and Schaefer, Štefankovič independently proved in 2002 that recognition of STRING graphs is decidable. Schaefer,Sedgwick and Štefankovič (2003) proved that recognizing of STRING graphs is NP-complete.
36 Consequence of STRING graphs STRING graphs are ISLAND graphs.
37 Consequence of STRING graphs STRING graphs are ISLAND graphs.
38 Consequence of STRING graphs STRING graphs are ISLAND graphs. There exist graphs which require exponential number of grid points.
39 Consequence of STRING graphs STRING graphs are ISLAND graphs. There exist graphs which require exponential number of grid points. The recognition of ISLAND graphs is NP-complete.
40 Problem The complexity of recognition of k-island graph.
41 Problem The complexity of recognition of k-island graph. We show that k-island is NP-complete for k < 3 and k > 5.
42 Our approach We use different method for the cases k < 3 and k > 5
43 Our approach We use different method for the cases k < 3 and k > 5 For cases k = 1,2 we made reduction from NAE-3-SAT and use so called logic engine.
44 Our approach We use different method for the cases k < 3 and k > 5 For cases k = 1,2 we made reduction from NAE-3-SAT and use so called logic engine. For cases k > 5 we use reduction from PLANAR-3-CONNECTED-(3,4)-SAT.
45 Logic engine The logic engine was designed by Eades and Whitesides.
46 Logic engine The logic engine was designed by Eades and Whitesides. Very intuitive model of a standard reduction from the problem NAE-SAT
47 Small islands Theorem The problem 1-ISLAND is NP-complete.
48 Small islands Theorem The problem 1-ISLAND is NP-complete. Theorem The problem 2-ISLAND is NP-complete.
49 How to prove it For logic engine construction are important to produce: rigid element (frames, shafts) flexible element (rod)
50 Case k = 1 rigid element a b x A B X d c y D C Y x y z X Y Z flexible element a b A B x c y X C Y a b A B
51 Case k = 2 rigid element a b A B flexible element a a A B B B b b
52 Example k = 1 ( X 1 X 2 X 3 ) ( X 1 X 2 X 4 ) ( X 2 X 3 X 4 )
53 Example k = 1 ( X 1 X 2 X 3 ) ( X 1 X 2 X 4 ) ( X 2 X 3 X 4 ) a4 a3 a2 a1 c3 c2 c1 f c 1 c 2 c 3
54 Example k = 1 ( X 1 X 2 X 3 ) ( X 1 X 2 X 4 ) ( X 2 X 3 X 4 ) a4 a3 a2 a1 c3 f c2 c1 c 1 c 2 c 3 X 1 false X 2 false X 3 true X 4 true
55 Example k = 2 ( X 1 X 2 X 3 ) ( X 1 X 2 X 4 ) ( X 2 X 3 X 4 ) f = a5 a4 a3 a2 a1 X 1 false X 2 false X 3 true X 4 true
56 PLANAR-3-CONNECTED-(3,4)-SAT A variant of satisfiability problem.
57 PLANAR-3-CONNECTED-(3,4)-SAT A variant of satisfiability problem. Input formula has exactly 3 distinct literals in each clause.
58 PLANAR-3-CONNECTED-(3,4)-SAT A variant of satisfiability problem. Input formula has exactly 3 distinct literals in each clause. Each variable occurs in at most 4 clauses
59 PLANAR-3-CONNECTED-(3,4)-SAT A variant of satisfiability problem. Input formula has exactly 3 distinct literals in each clause. Each variable occurs in at most 4 clauses Incidence graph of formula is vertex-3-connected and planar.
60 Reduction for large islands PLANAR-3-CONNECTED-(3,4)-SAT was introduced by by Kratochvíl. It can be used to show that STRING graphs are NP-hard.
61 Reduction for large islands PLANAR-3-CONNECTED-(3,4)-SAT was introduced by by Kratochvíl. It can be used to show that STRING graphs are NP-hard. We will use the same reduction for recognition of large islands.
62 Idea of the proof For given φ formula produce a graph H φ with following properties:
63 Idea of the proof For given φ formula produce a graph H φ with following properties: If φ is satisfiable then Hφ 6 ISLAND.
64 Idea of the proof For given φ formula produce a graph H φ with following properties: If φ is satisfiable then Hφ 6 ISLAND. If φ is not satisfiable then Hφ / STRING = ISLAND.
65 The first step of construction We fix rectilinear drawing of incidence graph of φ.
66 The first step of construction We fix rectilinear drawing of incidence graph of φ. Clauses and variables are located in points of planar grid.
67 The first step of construction We fix rectilinear drawing of incidence graph of φ. Clauses and variables are located in points of planar grid. Edges piece-wise linear and following the grid lines.
68 The first step of construction continued Consider a refinement of grid so that the variable and clause are replaced by disjoint rectangles
69 The first step of construction continued Consider a refinement of grid so that the variable and clause are replaced by disjoint rectangles Edges replaced by pair of parallel paths.
70 The first step of construction continued Consider a refinement of grid so that the variable and clause are replaced by disjoint rectangles Edges replaced by pair of parallel paths.
71 Variable gadget
72 Clause gadget
73 Representations of the variable gadget
74 Representations of the clause gadget
75 The end Thank you for your attention
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