Stickiness, Edge-Thickness, and Clique-Thickness in Graphs

In this paper we study the edge-thickness and the clique-thickness of a graph. The edge-thickness of a graph is defined as the thickness of the family of edges. The clique-thickness of a graph is defined as the thickness of the family of maximal cliques. Edges and maximal cliques of a graph are both considered as a collection of subsets of the vertex set. On one hand, we introduce a new parameter called stickiness. We show the relation between stickiness and edge-thickness, and show how the stickiness of a graph can be computed in a more efficient way. On the other hand, we show that the clique-thickness is equal to the reciprocal of the fractional stability number. For graphs having the property that the clique cover number is equal to the independence number it follows that the clique-thickness is equal to the reciprocal of the independence number, and in this case it is computable in polynomial time.

[1]  Maciej M. Sysło,et al.  On Hallian digraphs, permanents and transversals , 1979 .

[2]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[3]  William H. Cunningham.  Computing the binding number of a graph , 1990, Discret. Appl. Math..

[4]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[5]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[6]  D. R. Woodall.  The binding number of a graph and its Anderson number , 1973 .

[7]  Vlastimil Pták,et al.  A measure of thickness for families of sets , 1986, Discret. Math..

[8]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[9]  M. Rosenfeld.  Graphs with a large capacity , 1970 .

[10]  Danuta Michalak,et al.  On the binding number of some Hallian graphs , 1987 .

[11]  Danuta Michalak.  The Binding Number of k-Trees , 1992 .

[12]  Michel Minoux,et al.  Graphs and Algorithms , 1984 .

[13]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[14]  I. Anderson.  Perfect matchings of a graph , 1971 .

[15]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[16]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[17]  J. Seidel.  Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3 , 1968 .

[18]  Farid Alizadeh,et al.  A sublinear-time randomized parallel algorithm for the maximum clique problem in perfect graphs , 1991, SODA '91.

[19]  Fred S. Roberts,et al.  Analogues of the Shannon Capacity of a Graph , 1982 .

[20]  Jaroslav Nešetřil,et al.  Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity , 1992 .

[21]  M. Rosenfeld.  ON A PROBLEM OF C. E. SHANNON IN GRAPH THEORY , 1967 .

[22]  Mieczyslaw Borowiecki.  The Binding Number of Graphs , 1992 .

[23]  Ernst G. Straus,et al.  Which rational numbers are binding numbers? , 1981, J. Graph Theory.

[24]  Jan van Leeuwen,et al.  Graph Algorithms , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[25]  L. Lovász,et al.  Polynomial Algorithms for Perfect Graphs , 1984 .

[26]  Vlastimil Pták,et al.  Thickness of families of sets and a minimax lemma , 1992, Discret. Math..