Characterizing minimal interval completions: Background Motivation - PowerPoint PPT Presentation

Pinar Heggernes, Karol Suchan, Ioan Todinca, and Yngve Villanger Characterizing minimal interval completions: Background Motivation Towards better understanding of profile and Interval graphs Minimum and minimal interval pathwidth completion Defining the Characterization Folding A single edge Path- Pinar Heggernes, Karol Suchan, Ioan Todinca, and decomposition Defining graphs Yngve Villanger Defining foldings Algorithm FillFolding No single fill edge Pivots One or two unfolding STACS 2007, Aachen Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Outline Background 1 Motivation Pinar Heggernes, Karol Suchan, Ioan Interval graphs Todinca, and Yngve Villanger Minimum and minimal interval completion Background Defining the Characterization Motivation Interval graphs Folding 2 Minimum and minimal interval A single edge completion Defining the Path-decomposition Characterization Defining graphs Folding A single edge Defining foldings Path- decomposition Algorithm FillFolding Defining graphs Defining foldings No single fill edge Algorithm FillFolding Pivots No single fill edge Pivots One or two unfolding One or two unfolding Unfolding 3 Unfolding Recognizing a one unfolding Recognizing a one unfolding Recognizing a two unfolding Recognizing a two unfolding Extracting a minimal interval completion Extracting a minimal interval completion

Motivation Pinar Heggernes, Problems: minimum fill-in and tree-width Karol Suchan, Ioan Todinca, and Both problems are NP-hard Yngve Villanger The solution can be found among the Minimal triangulations Background Motivation Characterizations of minimal triangulations can be used to Interval graphs Minimum and bound the search space. minimal interval completion Defining the Several characterizations exist. Characterization Folding A single edge Path- Problems: profile and path-width decomposition Defining graphs Both problems are NP-hard Defining foldings Algorithm FillFolding The solution can be found among the Minimal interval No single fill edge Pivots completions One or two unfolding We will now have a look at the first characterization of a Unfolding minimal interval completion. Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Interval graphs Definition Pinar Heggernes, A graph is an interval graph if every vertex can be assigned Karol Suchan, Ioan Todinca, and an interval on the real line, such that two lines only intersect if Yngve Villanger the corresponding vertices are adjacent. Background Motivation An interval graph H = ( V, E ∪ F ) where E ∩ F = ∅ is called Interval graphs Minimum and an interval completion of G = ( V, E ) if E ⊆ F . minimal interval completion The edge set F is called fill-edges . Defining the Characterization Folding A single edge Path- decomposition Defining graphs Defining foldings Algorithm FillFolding No single fill edge Pivots One or two unfolding Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Minimum and minimal interval completion Definitions Pinar Heggernes, An interval graph H = ( V, E ∪ F ) is a minimum interval Karol Suchan, Ioan Todinca, and completion of G = ( V, E ) if E ∩ F = ∅ and H ′ = ( V, E ∪ F ′ ) Yngve Villanger is not an interval graph for every edge set F ′ such that Background | F ′ | < | F | . Motivation Interval graphs Minimum and An interval graph H = ( V, E ∪ F ) is a minimal interval minimal interval completion of G = ( V, E ) if E ∩ F = ∅ , and H ′ = ( V, E ∪ F ′ ) completion Defining the is not an interval graph for every edge set F ′ such that F ′ ⊂ F . Characterization Folding A single edge Path- decomposition Defining graphs Defining foldings Algorithm FillFolding No single fill edge Pivots One or two unfolding Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Defining the Characterization Pinar Heggernes, Karol Suchan, Ioan The questions Todinca, and Yngve Villanger Given a graph G = ( V, E ) and Background an interval completion H = ( V, E ∪ F ) of G where E ∩ F = ∅ . Motivation Interval graphs Minimum and We answer the following question in polynomial time: minimal interval completion Defining the Is H a minimal interval completion of G ? Characterization Folding Alternatively A single edge Do there exist an edge set F ′ ⊂ F such that H ′ = ( V, E ∪ F ′ ) Path- decomposition Defining graphs is an interval graph? Defining foldings Algorithm FillFolding Finally No single fill edge If H is not a minimal interval completion, can an edge set F ′ , Pivots One or two such that F ′ ⊂ F and H ′ = ( V, E ∪ F ′ ) is an interval graph be unfolding Unfolding found? Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Outline Background 1 Motivation Pinar Heggernes, Karol Suchan, Ioan Interval graphs Todinca, and Yngve Villanger Minimum and minimal interval completion Background Defining the Characterization Motivation Interval graphs Folding 2 Minimum and minimal interval A single edge completion Defining the Path-decomposition Characterization Defining graphs Folding A single edge Defining foldings Path- decomposition Algorithm FillFolding Defining graphs Defining foldings No single fill edge Algorithm FillFolding Pivots No single fill edge Pivots One or two unfolding One or two unfolding Unfolding 3 Unfolding Recognizing a one unfolding Recognizing a one unfolding Recognizing a two unfolding Recognizing a two unfolding Extracting a minimal interval completion Extracting a minimal interval completion

Removing a single edge First step Pinar Heggernes, If there exists a single edge e ∈ F such that H ′ = ( V, E ∪ F \ { e } ) Karol Suchan, Ioan Todinca, and is an interval graph, then we have the answer. Yngve Villanger Background Observation Motivation Interval graphs Removing a single edges will not always be sufficient. Minimum and minimal interval completion Defining the Characterization Folding A single edge Path- decomposition Defining graphs Defining foldings Algorithm FillFolding No single fill edge Pivots One or two unfolding Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Path-decomposition Pinar Heggernes, Definition Karol Suchan, Ioan Todinca, and Yngve Villanger A path-decomposition of a graph G = ( V, E ) is a sequence P = ( X 1 , X 2 , ..., X r ) of subsets of V called bags, such that Background Motivation Each vertex u ∈ V appears in some bag X i , Interval graphs Minimum and minimal interval For every edge xy ∈ E some bag X i contains both x and y . completion Defining the The set of bags that contains a vertex x appears consecutively Characterization Folding in P . A single edge Path- decomposition Defining graphs Defining foldings Definition Algorithm FillFolding A path decomposition P is called a clique path of the given graph No single fill edge Pivots G if the vertices in every bag induces a maximal clique in G . One or two unfolding There exists a clique path P of a graph G if and only if G is an Unfolding interval graph. Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Defining some graphs Pinar Heggernes, Karol Suchan, Ioan Todinca, and Graphs Yngve Villanger We need some graphs to work width: Background Motivation Let G = ( V , E ) be an arbitrary graph. Interval graphs Minimum and minimal interval Let H 2 = ( V , E ∪ F 2 ) be an interval completion of G completion Defining the ( E ∩ F 2 = ∅ ). Characterization Folding Let H 0 = ( V , E ∪ F 0 ) be a minimal interval completion of G A single edge ( E ∩ F 0 = ∅ and F 0 ⊂ F 2 ). Path- decomposition Defining graphs Let H 1 = ( V , E ∪ F 1 ) be an interval completion of G , where Defining foldings Algorithm F 0 ⊂ F 1 ⊂ F 2 . FillFolding No single fill edge Pivots By slightly abusing notation, we have: One or two unfolding G ⊂ H 0 ⊆ H 1 ⊂ H 2 . Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion

Defining foldings Definition of folding Pinar Heggernes, Let H be an interval graph, let Q be any permutation of the set of Karol Suchan, Ioan Todinca, and maximal cliques of H , and let P be a clique path of H . We say that Yngve Villanger ( H, Q, P ) is a folding of H . Background Motivation Interval graphs Minimum and minimal interval completion Defining the Characterization Folding A single edge Path- decomposition Defining graphs Defining foldings Algorithm FillFolding No single fill edge Pivots One or two unfolding Unfolding Recognizing a one unfolding Recognizing a two unfolding Extracting a minimal interval completion