Domination in circle graphs Nicolas Bousquet Daniel Gon calves - PowerPoint PPT Presentation

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´ ephan Thomass´ e Agape 2012 Domination in circle graphs

Circles graphs Dominating set Some positive results Open Problems Circles graphs 1 Dominating set 2 Some positive results 3 Open Problems 4 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 ����� ����� ��� ��� ���� ��� ���� ��� 4 ����� ����� ����� ����� ��� ��� ��� ���� ��� ���� 3 �� �� 2 ����� ����� ������� ������� ����� ����� ���� ���� ��� ��� 5 �� �� 4 ����� ������� ������� ����� ����� ����� ��� ��� 2 ������� ������� ����� ����� ��� ��� 6 ������� ������� ����� ����� ��� ��� ����� ������� ������� ����� ��� ��� 1 ������ ������ �������� �������� � � ����� ������� ����� ������� ��� ��� � 1 �������� ������ ������ �������� � � ��� ��� ����� ����� �� �� 5 7 ������ ������ ��� ��� ����� ����� �� �� ������ ������ ��� ��� ����� ����� �� �� ������ ������ �� �� ��� ��� ����� ����� �� �� �� �� ������ ������ 7 6 �� �� ����� ����� Domination in circle graphs

Circles graphs Dominating set Some positive results Open Problems Dominating set 4 3 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. Domination in circle graphs

Circles graphs Dominating set Some positive results Open Problems Dominating set 4 3 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

Circles graphs Dominating set Some positive results Open Problems Dominating set 4 3 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 4 3 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 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]-difficulty 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 Circles graphs 1 Dominating set 2 Some positive results 3 Open Problems 4 Domination in circle graphs

Circles graphs Dominating set Some positive results Open Problems Theorem (B., Gon¸ calves, Mertzios, Paul, Sau, Thomass´ e) Dominating set parameterized by the size of the solution is W [1]-hard. 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