Overlaying a Hypergraph with a Graph with Bounded Maximum Degree

Abstract

Let G and H be respectively a graph and a hypergraph defined on a same set of vertices, and let F be a fixed graph. We say that G F -overlays a hyperedge S of H if F is a spanning subgraph of the subgraph of G induced by S, and that it F -overlays H if it F -overlays every hyperedge of H. Motivated by structural biology, we study the computational complexity of two problems. The first problem, (∆ ≤ k) F -Overlay, consists in deciding whether there is a graph with maximum degree at most k that F -overlays a given hypergraph H. It is a particular case of the second problem Max (∆ ≤ k) F -Overlay, which takes a hypergraph H and an integer s as input, and consists in deciding whether there is a graph with maximum degree at most k that F -overlays at least s hyperedges of H. We give a complete polynomial/N P-complete dichotomy for the Max (∆ ≤ k)-F -Overlay problems depending on the pairs (F, k), and establish the complexity of (∆ ≤ k) F -Overlay for many pairs (F, k).

A Novel Hypercube-based Approach to Overlay Design Algorithms on Topic Distribution Networks

“…By utilizing the localization aware dissemination, the notification delay was reduced by 30% and the reconfiguration cost was reduced by 80%. On the other hand, to decrease the average node degree, Havet et al [30] performed a study on forming a hypergraph by following the node clustering approach.…”

“…While connecting the nodes, they did not consider node's topic interest and its contribution with other nodes based on their subscriptions. In addition to Havet et al [30]'s approach, the current study benefits from the topic interest which helps to decrease the average node degree dramatically.…”