Stephane Durocher 1 Debajyoti Mondal 1 Md. Saidur Rahman 2 1 Department of Computer Science, University of Manitoba 2 Department of Computer Science & Engineering, Bangladesh University of Engineering and Technology (BUET)

Pairwise Compatible Graphs: Easy to Construct February 14-16WALCOM T A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] d min = 4 d max = 7 a b c d e f h i j k

Pairwise Compatible Graphs: Easy to Construct February 14-16WALCOM a b c d e T 4 3 A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] d min = 4 d max = 7 1 f h i j k e / d / c / b / a / 5 G 5 ∈ [4,7]

Pairwise Compatible Graphs: Easy to Construct February 14-16WALCOM a b c d e T 4 3 A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] d min = 4 d max = 7 1 f h i j k e / d / c / b / a / 5 G 3 3 ∉ [4,7]

Pairwise Compatible Graphs: Easy to Construct February 14-16WALCOM a b c d e T 4 3 A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] d min = 4 d max = 7 1 h i j k e / d / c / b / a / ∈ [4,7] G f

Pairwise Compatible Graphs: Easy to Construct February 14-16WALCOM A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] a b c d e T 4 3 d min = 4 d max = 7 1 f h i j k e / d / c / b / a / How hard is the Reverse Problem? G

a b c d e T 2 3 d min = 4 d max = 7 1 f h i j k Pairwise Compatible Graphs: How to Recognize? February 14-16WALCOM A pairwise compatible graph of ( T, d min, d max )  has n vertices that correspond to the n leaves of T, and  two vertices are adjacent in G if and only if their tree distance is in [ d min, d max ] e / d / c / b / a / G Given a graph G, is there an edge weighted tree T and two non-negative integers d min, d max, such that G is a PCG of ( T, d min, d max )? (T, d min, d max ) ?

Recognizing PCG by Constructing ( T, d min, d max ) February 14-16WALCOM d min = 2 d max = 3.5 d min = 1.5 d max = 2 Input Output Another possible Output Given a graph G, is there an edge weighted tree T and two non-negative integers d min, d max, such that G is a PCG of ( T, d min, d max )?

Pairwise Compatibility Tree Construction February 14-16WALCOM Complete Graphs G Pairwise-Compatibility Tree of G for d min =2, d max = Pairwise-Compatibility Tree of G for d min =3, d max =3 Complete Bipartite Graphs G

Pairwise Compatibility Tree Construction February 14-16WALCOM r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r d min = 3r d max = 3r Trees

Historical Background February 14-16WALCOM [Kearney, Munro, Phillips 2003] PCGs are used to model evolutionary relationships among a set of organisms Evolutionary Tree Relation Graph

Historical Background February 14-16WALCOM [Kearney, Munro, Phillips 2003] PCGs are used to model evolutionary relationships among a set of organisms. [Kearney, Munro, Phillips 2003] Given a graph G, if we can construct ( T, d min, d max ) in polynomial time, then we can find a maximum clique of G in polynomial time. [Kearney, Munro, Phillips 2003] Conjecture: All graphs are PCG. [Yanhaona, Bayzid, Rahman 2010] Refuted the Conjecture showing a class of bipartite graphs with n > 14 that are not PCGs. a1a1 a2a2 a3a3 a4a4 a5a5 b1b1 b2b2 b3b3 b4b4 b5b5 b6b6 b7b7 b8b8 b9b9 b () = 10

Historical Background February 14-16WALCOM [Kearney, Munro, Phillips 2003] PCGs are used to model evolutionary relationships among a set of organisms. [Kearney, Munro, Phillips 2003] Given a graph G, if we can construct ( T, d min, d max ) in polynomial time, then we can find a maximum clique of G in polynomial time. [Kearney, Munro, Phillips 2003] Conjecture: All graphs are PCG. [Yanhaona, Bayzid, Rahman 2010] Refuted the Conjecture showing a class of bipartite graphs with n > 14 that are not PCGs. a1a1 a2a2 a3a3 a4a4 a5a5 b1b1 b2b2 b3b3 b4b4 b5b5 b6b6 b7b7 b8b8 b9b9 b () = 10 Not a PCG.

Historical Background February 14-16WALCOM [Kearney, Munro, Phillips 2003] PCGs are used to model evolutionary relationships among a set of organisms. [Kearney, Munro, Phillips 2003] Given a graph G, if we can construct ( T, d min, d max ) in polynomial time, then we can find a maximum clique of G in polynomial time. [Kearney, Munro, Phillips 2003] Conjecture: All graphs are PCG. [Yanhaona, Bayzid, Rahman 2010] Refuted the Conjecture showing a class of bipartite graphs with n > 14 that are not PCGs. [Calamoneri, Petreschi, Sinaimeri 2012] A subclass of split matrogenic graphs are PCGs. [Salma, Rahman 2012] Triangle free maximum degree three outerplanar graphs are PCGs. [Calamoneri, Frascaria, Sinaimeri, 2012] All graphs with at most seven vertices are PCGs.

Some Natural Questions February 14-16WALCOM What is the smallest graph that is not a PCG? PCG Recognition Problem: Given a graph G, can we determine whether G is a PCG in polynomial time? Our Contribution Is there a planar graph that is not a PCG? There exists a graph with eight vertices that is not a PCG. A variant of the PCG Recognition Problem is NP-hard. There exists a planar graph with sixteen vertices that is not a PCG

Two Useful Lemmas February 14-16WALCOM u v x T uv P uv is the longest path in T uvw Then either d T (w,x)≤ d T (u,x), or, d T (w,x)≤ d T (v,x). w w Lemma 1: [Yanhaona et al. 2010] Let P uv be the longest path in T uvw, where u,v,w are leaves of T. Then for any leaf x ∉ {u,v,w} of T, either d T (w,x)≤ d T (u,x), or, d T (w,x)≤ d T (v,x). Lemma 1 x d min = 2 d max = 3.5 Input Output

Two Useful Lemmas February 14-16WALCOM a b d T uv P uv is the longest path in T uvw Then either d T (w,x)≤ d T (u,x), or, d T (w,x)≤ d T (v,x). c Lemma 1 Lemma 2: (This presentation.) Let G is a PCG of (T, d min, d max ), where G is a cycle a /,b /,c /,d /. Then d T (a,c) and d T (b,d) cannot be both greater than d max. Lemma 2 a/a/ b/b/ c/c/ d/d/ d min d max T Then either d T (a,c) < d min, or, d T (b,d) < d min. u v x w x w

Proof of Lemma 2 February 14-16WALCOM a b d c Lemma 2 a/a/ b/b/ c/c/ d/d/ d min d max T Then either d T (a,c) < d min, or, d T (b,d) < d min. a b c d Proof. W.l.o.g., assume that T does not have any vertex of degree two. a b c d x y x+y p q p+q T T G

Proof of Lemma 2 February 14-16WALCOM a b d c Lemma 2 a/a/ b/b/ c/c/ d/d/ d min d max T Then either d T (a,c) < d min, or, d T (b,d) < d min. Proof. We only need to consider the following tree topologies. If both d T (a,c) and d T (b,d) are greater than d max, then x+2y+z+p+q > 2d max. Since a / and d / are adjacent, d min ≤ x+y+q ≤ d max Since b / and c / are adjacent, d min ≤ p+y+z ≤ d max x+2y+z+p+q ≤ 2d max. G

Proof of Lemma 2 February 14-16WALCOM a b d c Lemma 2 a/a/ b/b/ c/c/ d/d/ d min d max T Then either d T (a,c) < d min, or, d T (b,d) < d min. Proof. We only need to consider the following tree topologies. If both d T (a,c) and d T (b,d) are greater than d max, then x+2y+z+p+q > 2d max. Since a / and d / are adjacent, d min ≤ x+y+q ≤ d max Since b / and c / are adjacent, d min ≤ p+y+z ≤ d max x+2y+z+p+q ≤ 2d max. If both d T (a,c) and d T (b,d) are greater than d max, then x+z > d max and p+q > d max. If x+p > d max, then a /,b / are non adjacent in G. If x+q > d max, then a /,d / are non adjacent in G. If z+p > d max, then b /,c / are non adjacent in G. If z+q > d max, then c /,d / are non adjacent in G. One of these four conditions must be true. Therefore, G cannot be a cycle. Contradiction! G

A Graph with Eight Vertices that is Not a PCG February 14-16WALCOM a b d T uv P uv is the longest path in T uvw Then either d T (w,x)≤ d T (u,x), or, d T (w,x)≤ d T (v,x). c w Lemma 1 Lemma 2 a/a/ b/b/ c/c/ d/d/ d min d max T Then either d T (a,c) < d min, or, d T (b,d) < d min. u v x w G Lemma 1 Lemma 2 G / G / is not a PCG. x

A Graph with Eight Vertices that is Not a PCG February 14-16WALCOM Lemma 1 Lemma 2 G / G / is not a PCG. Let G be a graph determined by exactly two edges (u /,v / ) and (w /,x / ). Then at least one of d T (u, w), d T (u, x), d T (v, w) and d T (v, x) must be greater than d max. a/a/ b/b/ c/c/ d/d/ e/e/ f / g/g/ h/h/ Proof. G u/u/ v/v/ w/w/ x/x/ u v y x d min d max T Suppose for a contradiction that all those four distances are less than d min. By Lemma 1, either d T (w,x) ≤ d T (u,x) or, d T (w,x) ≤ d T (v,x) u v w Then P uv is the longest path in T uvw. < d min Therefore, w / and x / cannot be adjacent! x

A Graph with Eight Vertices that is Not a PCG February 14-16WALCOM Lemma 1 Lemma 2 G / G / is not a PCG. Let G be a graph determined by exactly two edges (u /,v / ) and (w /,x / ). Then at least one of d T (u, w), d T (u, x), d T (v, w) and d T (v, x) must be greater than d max. a/a/ b/b/ c/c/ d/d/ e/e/ f / g/g/ h/h/ Proof. c/c/ d/d/ g/g/ h / b/b/ e/e/ f / a/a/ Applying this argument on G /, we observe that  at least one of d T (a,e), d T (a, f ), d T (b,e), d T (b,f) is greater than d max, and  at least one of d T (c,g), d T (c, h ), d T (d,g), d T (d,h) is greater than d max. Without loss of generality assume that  d T (a, f ) > d max, and  d T (d,g)> d max. a/a/ d/d/ f / g/g/ But a /,f / and d /,g / are non-adjacent. Hence by Lemma 2, both d T (a, f ) and d T (d,g) cannot be greater than d max.

A Planar Graph that is Not a PCG February 14-16WALCOM b/b/ c/c/ d / a/a/ At least one of d T (a,c), d T (a, d ), d T (b,c), d T (b,d) is greater than d max. Lemma 3. The tree distance between the degree four vertices of H must be less than d min. Proof. Skip. H x / y / G a / d / a / c / b / c / b / d / i / n / o / p / j / k / l / m/m/ e / f / g / h / q / r / s / t / By Lemma 3, all of d T (a,c), d T (a, d ), d T (b,c), d T (b,d) must be smaller than d min.

A Planar Graph that is Not a PCG February 14-16WALCOM G G has 20 vertices, but can we find such a planar graph with smaller number of vertices?

A Planar Graph with 16 Vertices that is Not a PCG February 14-16WALCOM Yes! Our proof holds even if we merge (i /,h / ), (l /,m / ), (p /,q / ) and (t /,e / ). G has 20 vertices, but can we find such a planar graph with smaller number of vertices? G 16 G Smallest Example? Open

Time Complexity of the PCG Recognition Problem? February 14-16WALCOM Problem : Max-Generalized-PCG-Recognition Instance : A graph G, a subset S of the edges of its complement graph, and a positive integer k. Question : Is there a PCG G / = PCG ( T, d min, d max ) such that G / contains G as a subgraph, but does not contain any edge of S ; and at least k edges of have distance greater than d max between their corresponding leaves in T ? Problem : Generalized-PCG-Recognition Instance : A graph G, a subset S of the edges of its complement graph. Question : Is there a PCG G / = PCG ( T, d min, d max ) such that G / contains G as a subgraph, but does not contain any edge of S ? Note that if S contains all the edges of the complement graph, then G is a PCG. We only prove the following variant is NP-hard. NP-hard ? Open

Reduction from Monotone-One-In-Three 3-SAT February 14-16WALCOM Problem : Max-Generalized-PCG-Recognition Instance : A graph G, a subset S of the edges of its complement graph, and a positive integer k. Question : Is there a PCG G / = PCG ( T, d min, d max ) such that G / contains G as a subgraph, but does not contain any edge of S ; and at least k edges of have distance greater than d max between their corresponding leaves in T ? Problem : Monotone-One-In-Three-3-SAT Instance : A set U of variables and a collection C of clauses over U such that each clause consists of exactly three non-negated literals. Question : Is there a satisfying truth assignment for U such that each clause in C contains exactly one true literal?

Idea of Reduction February 14-16WALCOM A NOT gate. In any pairwise compatibility tree, if a /,b / has tree distance greater than d max, then c /,d / has tree distance smaller than d min and vice versa.

Idea of Reduction February 14-16WALCOM N1N1 N2N2 N3N3 For each clause, build this graph such that no two literals can be realized as True, i.e., tree distance greater than d max. For example, if a / b / and c / d / both are realized by tree distance greater than d max, then by Lemma 2, both of the inputs of N 1 are realized by tree distance smaller than d min, which violates the property of a not gate.

Idea of Reduction February 14-16WALCOM For a variable that appears in different clauses, make such a chain such that their truth values are consistent. For example, here c / d / and k / l / are representing the same literal. Therefore, if c / d / is realized by tree distance greater than d max, m / n / is realized by tree distance less than d min, and hence k / l / and has the same tree distance realization as c / d /.

 Does there exist a split matrogenic graph that is not a PCG?  What is the smallest planar graph that is not a PCG?  Does there exist an outer planar graph that is not a PCG?  What is the complexity of (generalized) PCG recognition? February 14-16WALCOM Open Problems