On support and recognition problems for sparse hypergraphs

Abstract

A hypergraph H is a pair (V, E), where V is a set of vertices, and E is a collection of subsets of V , called hyperedges. They are used to express complex relations, and they generalize graphs where each element of E is a 2-element subset of V . Hypergraphs are one of the most important combinatorial objects of study in theoretical computer science, and have applications in several domains, including network design, scheduling problems, biology, machine learning, etc. Thus, it is important to study their structural properties. Starting with the work of Zykov [Zyk74], Voloshina and Feinberg [VF84], and John- son and Pollack [JP87], researchers have made several attempts to study the structure of a hypergraph by associating with it an appropriate graph. While their initial attempts were to introduce the planarity of a hypergraph, the notion developed in [VF84; JP87] can be generalized and is now called a support. A support for a hypergraph H = (V, E) is a graph Q = (V, F ) such that for each hyperedge E ∈ E, the induced subgraph Q[E] on the elements of E is connected. With this notion, a hypergraph is considered planar if it admits a support that is a planar graph. The concept of support has practical applications in hypergraph visualization, net- work design, and several optimization problems. Although deciding whether a hyper- graph admits a planar support is NP-hard, identifying sufficient conditions for the existence of such supports, particularly sparse or structured ones, remains a compelling research direction. Most of this thesis delves into the construction of supports for various graph classes. This thesis is divided into three parts. In Part (A), we consider hypergraphs defined by subgraphs of a given host graph. Let G = (V, E) be a graph and H be a collection of subgraphs of G. Then the pair (G, H) naturally defines a hypergraph with vertex set V and a hyperedge V (H) for each H ∈ H. We study support construction in three different settings, depending on whether the host graph G belongs to the class of graphs of (i) bounded genus, (ii) outerplanar, or (iii) bounded treewidth. We gave sufficient conditions that ensure the existence of a support from the same family of graphs as G. The results are extended to dual hypergraphs and to a more general setting- the intersection hypergraphs. We also present a fast algorithm for the construction of a planar support with straight-line embedding when the underlying hypergraph is defined by axis-parallel rectangles and points in R2. Part (B) of the thesis explores the role of supports in solving classical problems such as packing, covering, and coloring problems in hypergraphs. We study these problems for hypergraphs arising from subgraphs of a host graph as well as from geometric regions on orientable surfaces, and present approximation results to the packing and covering problems above. Finally, Part (C) turns to abstract hypergraphs and examines the computational complexity of identifying vertex orderings that forbid fixed patterns. We show NP- hardness of this problem for several vertex orderings, and we deduce implications for the recognition of hypergraphs defined by geometric regions in R2.