K-tree

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

Finding All Maximal Cliques of a Family of Induced Subgraphs

Abstract: Many real world problems can be mapped onto graphs and solved with well-established efficient algorithms studied in graph theory. One such problem is to find large sets of points satisfying some mutual relationship. This problem can be transformed to the problem of finding all cliques of an undirected graph by mapping each point onto a vertex of the graph and connecting any two vertices by an edge whose corresponding points satisfy our desired relationship. Clique detection has been widely studied and there exist efficient algorithms. In this paper we study a related problem, where all points have a set of binary attributes, each of which is either 0 or 1. This is only a small limitation, since all discrete properties can be mapped onto binary attributes. In our case, we want to find large sets of points not only satisfying some mutual relationship; but, in addition, all points of a set also need to have at least one common attribute with value 1. The problem we described can be mapped onto a set of induced subgraphs, where each subgraph represents a single attribute. For attribute $i$, its associated subgraph contains those vertices corresponding to the points with attribute $i$ set to 1. We introduce the notion of a maximal clique of a family, $\mathcal{G}$, of induced subgraphs of an undirected graph, and show that determining all maximal cliques of $\mathcal{G}$ solves our problem. Furthermore, we present an efficient algorithm to compute all maximal cliques of $\mathcal{G}$. The algorithm we propose is an extension of the widely used Bron-Kerbosch algorithm.

Vertex Partitioning of Crown-Free Interval Graphs

Abstract: We study the problem of finding an acyclic orientation of an undirected graph G such that each path is contained in a limited number of maximal cliques of G. In general, in an acyclic oriented graph, each path is contained in more than one maximal cliques. We focus our attention on crown-free interval graphs, and show how to find an acyclic orientation of such a graph, which guarantees that each path is contained in at most four maximal cliques. The proposed technique is used to find approximated solutions for a class of related optimization problems where a solution corresponds to an acyclic orientation of graphs.

Packing Tree Degree Sequences

Abstract: A degree sequence is a list of non-negative integers, $${D = d_1, d_2, \ldots , d_n}$$. It is called graphical if there exists a simple graph G such that the degree of the ith vertex is $$d_i$$; G is then said to be a realization of D. A tree degree sequence is one that is realized by a tree. In this paper we consider the problem of packing tree degree sequences: given k tree degree sequences, do they have simultaneous (i.e. on the same vertices) edge-disjoint realizations? We conjecture that this is true for any arbitrary number of tree degree sequences whenever they share no common leaves (degree-1 vertices). This conjecture is inspired by work of Kundu (SIAM J Appl Math 28:290–302, 1975) that showed it to be true for 2 and 3 tree degree sequences. In this paper, we give a proof for 4 tree degree sequences and a computer-aided proof for 5 tree degree sequences. We also make progress towards proving our conjecture for arbitrary k. We prove that k tree degree sequences without common leaves and at least $$2k-4$$ vertices which are not leaves in any of the trees always have edge-disjoint tree realizations. Additionally, we show that to prove the conjecture, it suffices to prove it for $$n \le 4k - 2$$ vertices. The main ingredient in all of the presented proofs is to find rainbow matchings in certain configurations.