Split Decomposition and Circle Graph Recognition in Quasi-Linear Time Christophe Paul CNRS - LIRMM - Universit´eMontpellier II, France March 25, 2009 Joint work with D. Corneil (U. of Toronto), E. Gioan (CNRS LIRMM) and M. Tedder (U. of Toronto) Graph-labeled tree (GLT) I a pair (T ; F) with T a tree and F a set of graphs I each node v of degree k of T is labelled by a graph Gv 2 F I a bijection ρv from the tree-edges incident to v to V (Gv ) 2 1 3 8 7 4 5 6 Accessiblity graph of a GLT Given a GLT (T ; F), define the graph G(T ; F) with I vertex set of G(T ; F) = set of leaves of T I xy is an edge iff ρv (uv)ρv (vw) 2 E(Gv ) for all tree-edges uv, vw on the x; y-path in T 2 1 2 1 3 8 8 4 3 7 7 4 5 6 5 6 A split in a graph G = (V ; E) is a bipartition (A; B) of V st. I jAj > 2, jBj > 2 and I for x 2 A, y 2 B, xy 2 E iff x 2 N(B) and y 2 N(A) 2 1 2 1 3 8 8 4 3 7 7 4 5 6 5 6 Every edge of a GLT defines a split 2 3 2 3 split 1 1 q r join N(r) N(q) 4 4 5 5 Theorem [Cunnigham'82 reformulated] For any connected graph G, there exists a unique graph-labelled tree ST (G) = (T ; F) with a minumun number of nodes such that I G = G(T ; F), I any graph of F is prime or degenerate. Prime Degenerate Vertex incremental characterization A vertex x is added to a graph G = (V ; E) with neighbors S ⊆ V . (S is represented in red) I How to compute the split tree ST (G + (x; S)) from ST (G)? I q is empty if L(q) \ S = ; I q is perfect if every leaf in L(q) \ S is accessible from q I q is mixed otherwise States of marker vertices Let q be marker vertex (or a leaf) Let L(q) be the set of leaves l such that the path between l and q contains the edge of the tree associated with q q I q is perfect if every leaf in L(q) \ S is accessible from q I q is mixed otherwise States of marker vertices Let q be marker vertex (or a leaf) Let L(q) be the set of leaves l such that the path between l and q contains the edge of the tree associated with q I q is empty if L(q) \ S = ; E E E E E E I q is mixed otherwise States of marker vertices Let q be marker vertex (or a leaf) Let L(q) be the set of leaves l such that the path between l and q contains the edge of the tree associated with q I q is empty if L(q) \ S = ; I q is perfect if every leaf in L(q) \ S is accessible from q P E P E E P E P E P E P States of marker vertices Let q be marker vertex (or a leaf) Let L(q) be the set of leaves l such that the path between l and q contains the edge of the tree associated with q I q is empty if L(q) \ S = ; I q is perfect if every leaf in L(q) \ S is accessible from q I q is mixed otherwise P E P E M M E P M E P M E P E P I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect P P I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect x PP I 3 - T has a fully-mixed node u Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect I 2 - T has an edge e, one extremity of which is perfect and the other empty x P E Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u every marker vertex adjacent to the node u is mixed M M M Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u (the set of fully-mixed nodes is a subtree of T ) M M Make a join M M E P M M E P M M Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u (the set of fully-mixed nodes is a subtree of T ) E P E P Vertex Incremental Characterization The split tree of G + (x; S) is obtained from ST (G) by 3 cases: I 1 - T has an edge e whose extremities are perfect I 2 - T has an edge e, one extremity of which is perfect and the other empty I 3 - T has a fully-mixed node u (the set of fully-mixed nodes is a subtree of T ) x Prime node unless x has a twin E P E P Theorem: the incremental split decomposition algorithm runs in O((n + m)α(n; m)) Ingredients for the complexity analysis I the vertex insertion ordering has to be a LexBFS ordering I a carefully amortizing analysing based on discharging techniques Previous results I [Cunningham 1982] O(nm) 2 I [Ma, Spinrad 1994] O(n ) I [Dahlhaus 2000] O(n + m) (tough !) I [Charbit, De Montgolfier, Raffinot 2009] O(n + m) (today) I A graph is circle iff all the prime nodes of its split tree are circle graphs I A circle graph is prime for the split decomposition iff it has a unique (up to mirror) realizer Application to circle graphs recognition I A circle graph is the intersection graph of chords in a circle. I A circle graph is prime for the split decomposition iff it has a unique (up to mirror) realizer Application to circle graphs recognition I A circle graph is the intersection graph of chords in a circle. I A graph is circle iff all the prime nodes of its split tree are circle graphs 2 1 2 1 3 3 8 8 7 7 4 4 5 6 5 6 Application to circle graphs recognition I A circle graph is the intersection graph of chords in a circle. I A graph is circle iff all the prime nodes of its split tree are circle graphs I A circle graph is prime for the split decomposition iff it has a unique (up to mirror) realizer Application to circle graphs recognition I A circle graph is the intersection graph of chords in a circle. I A graph is circle iff all the prime nodes of its split tree are circle graphs I A circle graph is prime for the split decomposition iff it has a unique (up to mirror) realizer Extreme chord - vertex A vertex x is extreme if its chord c(x) cut the realizer in a way that the chords of all the non-neighors of x are either all on the right or all on the left of c(x). First LexBFS Lemma Let G be a circle graph and σ be a LexBFS ordering of G ending at x. Then there exists a realizer of G in which the chord c(x) is extreme. 5 1 7 1 2 7 5 2 8 3 4 6 3 4 8 6 First LexBFS Lemma Let G be a circle graph and σ be a LexBFS ordering of G ending at x. Then there exists a realizer of G in which the chord c(x) is extreme. 5 1 7 1 2 7 5 2 8 3 4 6 3 4 8 6 I Consequence: we know in constant time where to insert x in a given realizer of G. I But many realizers unless prime A second LexBFS Lemma Let σ be a LexBFS ordering of G = GS (T ; F). The "induced" ordering σu of the marker vertices of Gu is a LexBFS ordering.