One-node cutsets and the dominating set polytope

In this paper we study a composition (decomposition) technique for the dominating set polytope in graphs which are decomposable by one-node cutsets. If G decomposes into G1 and G2, we show that the dominating set polytope of G can be described from two linear systems related to G1 and G2. This gives a way to characterize this polytope for classes of graphs that can be recursively decomposed. This also gives a procedure to describe facets for this polytope. Application of these techniques is discussed for the class of the cactus.

[1]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra III: Graphs with No W4 Minor , 1994, SIAM J. Discret. Math..

[2]  Jean Fonlupt,et al.  Polyhedral consequences of the amalgam operation , 1994, Discret. Math..

[3]  Michel Balinski,et al.  Of Stable Marriages and Graphs, and Strategy and Polytopes , 1997, SIAM Rev..

[4]  O. Ore.  Theory of Graphs , 1962 .

[5]  Martin Farber,et al.  Domination in Permutation Graphs , 1985, J. Algorithms.

[6]  Stephen T. Hedetniemi,et al.  Towards a theory of domination in graphs , 1977, Networks.

[7]  Claude Berge.  Theory of graphs and its applications , 1962 .

[8]  Tohru Kikuno,et al.  A linear algorithm for the domination number of a series-parallel graph , 1983, Discret. Appl. Math..

[9]  Uriel G. Rothblum,et al.  Characterization of stable matchings as extreme points of a polytope , 1992, Math. Program..

[10]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[11]  Ali Ridha Mahjoub,et al.  On a composition of independence systems by circuit identification , 1991, J. Comb. Theory, Ser. B.

[12]  D. R. Fulkerson,et al.  On balanced matrices , 1974 .

[13]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[14]  Jean Fonlupt,et al.  Compositions in the bipartite subgraph polytope , 1992, Discret. Math..

[15]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[16]  Claude Berge,et al.  The theory of graphs and its applications , 1962 .

[17]  Robert B. Allan,et al.  On domination and independent domination numbers of a graph , 1978, Discret. Math..

[18]  Ali Ridha Mahjoub,et al.  Polytope Des Absorbants Dans Une Classe De Graphe a Seuil , 1983 .

[19]  F. Barahona.  The max-cut problem on graphs not contractible to K5 , 1983 .

[20]  Maurizio A. BONUCCELLI.  Dominating sets and domatic number of circular arc graphs , 1985, Discret. Appl. Math..

[21]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra II: Stable Sets , 1994, SIAM J. Discret. Math..

[22]  Martin Farber,et al.  Domination, independent domination, and duality in strongly chordal graphs , 1984, Discret. Appl. Math..

[23]  C. L. Liu,et al.  Introduction to Combinatorial Mathematics. , 1971 .

[24]  Dieter Kratsch,et al.  Domination on Cocomparability Graphs , 1993, SIAM J. Discret. Math..

[25]  Alan A. Bertossi,et al.  Dominating Sets for Split and Bipartite Graphs , 1984, Inf. Process. Lett..

[26]  Stephen T. Hedetniemi,et al.  A linear algorithm for finding a minimum dominating set in a cactus , 1986, Discret. Appl. Math..

[27]  Lorna Stewart,et al.  Dominating sets in perfect graphs , 1991, Discret. Math..

[28]  F. Margot.  Composition de polytopes combinatoires : une approche par projection , 1995 .