K-tree

About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.

Polynomial-time solvability of the maximum clique problem

Abstract: The maximum clique problem is known to be a typical NP-complete problem, and hence it is believed to be impossible to solve it in polynomial-time. So, it is important to know a reasonable sufficient condition under which the maximum clique problem can be proved to be polynomial-time solvable. In this paper, given a graph of n vertices and whose maximum degree is Δ, we prove that if Δ is less than or equal to 2.493dlg n (d≥1: a constant), then the maximum clique problem is solvable in the polynomial time of O(n2+d). The proof is based on a very simple algorithm which is obtained from an algorithm CLIQUES that generates all maximal cliques in a depth-first way in O(3n/3)-time (which is published in Theoretical Computer Science 363, 2006, as "The worstcase time complexity for generating all maximal cliques and computational experiments" by E. Tomita et al.). The proof itself is very simple.

On the interval number of a chordal graph

Abstract: The interval number of a (simple, undirected) graph G is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t real intervals. A chordal (or triangulated) graph is one with no induced cycles on 4 or more vertices. If G is chordal and has maximum clique size ω(G) = m, then i(G) ⩽ [1 + o(1)]m/log2m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).

Flag Vectors of Multiplicial Polytopes

Abstract: Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes . These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the $h$-vector of the ordinary $d$-polytope with $n+1$ vertices and characteristic $k$: $h_i={k-d+i\choose i}+(n-k){k-d+i-1\choose i-1}$, for $i\le d/2$. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky.

Mining of Frequent Externally Extensible Outerplanar Graph Patterns

Abstract: An outerplanar graph is a planar graph which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many chemical compounds are known to be expressed by outerplanar graphs. In this paper, firstly, we introduce an externally extensible outerplanar graph pattern (eeo-graph pattern for short) as a graph pattern common to a finite set of outerplanar graphs like a dataset of chemical compounds. The eeo-graph pattern can express a substructure common to blocks which appear in outerplanar graph structured data. Secondly, we propose a data mining algorithm for enumerating all maximal frequent eeo-graph patterns from a finite set of outerplanar graphs. Finally, we report experimental results on a chemical dataset.