TY - GEN

T1 - What is between chordal and weakly chordal graphs?

AU - Cohen, Elad

AU - Golumbic, Martin Charles

AU - Lipshteyn, Marina

AU - Stern, Michal

PY - 2008

Y1 - 2008

N2 - An (h,s,t)-representation of a graph G consists of a collection of subtrees {S v | v ∈ V(G)} of a tree T, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [∞, ∞, 1] = [3,3,1] = [3,3,2] graphs (notation of ∞ here means that no restriction is imposed). We investigate the complete bipartite graph K 2,n and prove new theorems characterizing those K 2,n graphs that have an (h,s,2)-representation and those that have an (h,s,3)-representation. We characterize [3,2,4] graphs as equivalent to the 4-flower-free [2,4,4] graphs and give a recognition algorithm for [2,3,4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h,s,t] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h,s,t] framework.

AB - An (h,s,t)-representation of a graph G consists of a collection of subtrees {S v | v ∈ V(G)} of a tree T, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [∞, ∞, 1] = [3,3,1] = [3,3,2] graphs (notation of ∞ here means that no restriction is imposed). We investigate the complete bipartite graph K 2,n and prove new theorems characterizing those K 2,n graphs that have an (h,s,2)-representation and those that have an (h,s,3)-representation. We characterize [3,2,4] graphs as equivalent to the 4-flower-free [2,4,4] graphs and give a recognition algorithm for [2,3,4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h,s,t] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h,s,t] framework.

UR - http://www.scopus.com/inward/record.url?scp=58449103290&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-92248-3_25

DO - 10.1007/978-3-540-92248-3_25

M3 - Conference contribution

AN - SCOPUS:58449103290

SN - 3540922474

SN - 9783540922476

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 275

EP - 286

BT - Graph-Theoretic Concepts in Computer Science - 34th International Workshop, WG 2008, Revised Papers

T2 - 34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008

Y2 - 30 June 2008 through 2 July 2008

ER -