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Domination in Circle Graphs Circles graphs Dominating set Some positive results Open Problems Domination in circle graphs Nicolas Bousquet Daniel Gon¸calves George B. Mertzios Christophe Paul Ignasi Sau St´ephanThomass´e WG 2012 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Circle graphs Circle graph A circle graph is a graph which can be represented as an intersection graph of chords in a circle. 3 3 4 2 5 4 2 6 1 1 5 7 7 6 Domination in circle graphs Independent dominating sets. Connected dominating sets. Total dominating sets. All these problems are NP-complete. Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. Domination in circle graphs Connected dominating sets. Total dominating sets. All these problems are NP-complete. Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. Independent dominating sets. Domination in circle graphs Total dominating sets. All these problems are NP-complete. Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. Independent dominating sets. Connected dominating sets. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. Independent dominating sets. Connected dominating sets. Total dominating sets. All these problems are NP-complete. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Hardness of Dominating Set Dominating Set is polynomial on proper interval graphs. Dominating Set is polynomial on permutation graphs. Dominating Set is W [1]-hard on general graphs. Dominating Set is NP-complete on circle graphs. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs k-Colored Clique Input : G colored with k-colors. n vertices of each color. Parameter : k. Output : YES iff there is a clique of size k with one vertex of each color. Theorem k-Colored Clique is W [1]-hard parameterized by k. Circles graphs Dominating set Some positive results Open Problems Main result Theorem (B., Gon¸calves,Mertzios, Paul, Sau, Thomass´e) Dominating Set parameterized by the size of the solution is W [1]-hard on circle graphs. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Main result Theorem (B., Gon¸calves,Mertzios, Paul, Sau, Thomass´e) Dominating Set parameterized by the size of the solution is W [1]-hard on circle graphs. k-Colored Clique Input : G colored with k-colors. n vertices of each color. Parameter : k. Output : YES iff there is a clique of size k with one vertex of each color. Theorem k-Colored Clique is W [1]-hard parameterized by k. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Reduction from k-colored clique Idea : Simulate the behavior of the vertices of each color. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Reduction from k-colored clique Idea : Simulate the behavior of the vertices of each color. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Transformation into a circle graph 1 2 3 4 5 1 2 3 4 5 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Transformation into a circle graph 1 2 3 4 5 1 2 3 4 5 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Transformation into a circle graph 1 2 3 n 1 2 3 n Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Transformation into a circle graph 1 2 3 n 1 2 3 n Domination in circle graphs o Theorem Total Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Circles graphs Dominating set Some positive results Open Problems Corollaries Theorem Connected Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Domination in circle graphs o Theorem Total Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Circles graphs Dominating set Some positive results Open Problems Corollaries Theorem Connected Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Domination in circle graphs o Circles graphs Dominating set Some positive results Open Problems Corollaries Theorem Connected Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Theorem Total Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Domination in circle graphs Theorem Acyclic Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Circles graphs Dominating set Some positive results Open Problems Independent Dominating set Theorem Independent Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Independent Dominating set Theorem Independent Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Theorem Acyclic Dominating Set is W [1]-hard on circle graphs parameterized by the size of the solution. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs Theorem Input : A circle graph G, an integer k. Output : YES iff there exists a dominating path of length k. This problem is in P. Circles graphs Dominating set Some positive results Open Problems Some positive results Theorem Input : A circle graph G, an integer k. Output : YES iff there exists a dominating tree of size k. This problem is in P. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Some positive results Theorem Input : A circle graph G, an integer k. Output : YES iff there exists a dominating tree of size k. This problem is in P. Theorem Input : A circle graph G, an integer k. Output : YES iff there exists a dominating path of length k. This problem is in P. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems An FPT result Theorem Input : A circle graph G, a tree T of size k. Parameter : k Output : YES iff there exists a dominating tree isomorphic to T . This problem is in NP-complete and FPT. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs Dominating Set parameterized by treewidth is FPT. Dominating Set parameterized by treewidth does not have a polynomial kernel. Open problem Does the domination problem in circle graphs admits a polynomial kernel parameterized by treewidth ? Circles graphs Dominating set Some positive results Open Problems Open problems Conjecture Bounded Treewidth Connected Dominating Set is polynomial in circle graphs. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Open problems Conjecture Bounded Treewidth Connected Dominating Set is polynomial in circle graphs. Dominating Set parameterized by treewidth is FPT. Dominating Set parameterized by treewidth does not have a polynomial kernel. Open problem Does the domination problem in circle graphs admits a polynomial kernel parameterized by treewidth ? Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Thanks for your attention Any question ? Domination in circle graphs.
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