A classical theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs – a graph G is called dista...

In this paper we modify slightly Razborov’s flag algebras machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from √...

This paper studies the hierarchical broadcast networks, which are deened as broadcast networks whose every connected induced subgraph is also a broadcast network. We show that the minimum number of edges of a hierarchical broadcast network of order n is d n(n?2) 2 e for any n 4.

Abstract An e cient heuristic is presented for the problem of nding a minimum size k connected spanning subgraph of an undirected or directed simple graph G V E There are four versions of the problem and the approximation guarantees are as followsAn e cient heuristic is presented for the problem of nding a minimum size k connected spanning subgraph of an undirected or directed simple graph G V ...

Given a query graph q, a “subgraph-search” algorithm retrieves from a graph database D all graphs that have q as a subgraph, D(q). Subgraph search is costly because of its involvement of a subgraph-isomorphism test, which is a NPcomplete problem. Graph indexes are used to improve the algorithm efficiency by first filtering out a set of false answers and then verifying each graph that has passed...

Mining discriminative features for graph data has attracted much attention in recent years due to its important role in constructing graph classifiers, generating graph indices, etc. Most measurement of interestingness of discriminative subgraph features are defined on certain graphs, where the structure of graph objects are certain, and the binary edges within each graph represent the "presenc...

Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about a connected k-...

A graph is called biclaw-free if it has no biclaw as an induced subgraph. In this note, we prove that if G is a connected bipartite biclaw-free graph with δ(G) ≥ 5, then G is collapsible, and of course supereulerian. This bound is best possible.

The Erdös-Hajnal conjecture states that for every graph H, there exists a constant δ(H) > 0 such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least |V (G)|. This paper is a survey of some of the known results on this conjecture.