On Hypergraph Supports

by   Rajiv Raman, et al.

Let ℋ=(X,ℰ) be a hypergraph. A support is a graph Q on X such that for each E∈ℰ, the subgraph of Q induced on the elements in E is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph G=(V,E), with c:V→{𝐫,𝐛}, and a collection of connected subgraphs ℋ of G, a primal support is a graph Q on 𝐛(V) such that for each H∈ℋ, the induced subgraph Q[𝐛(H)] on vertices 𝐛(H)=H∩ c^-1(𝐛) is connected. A dual support is a graph Q^* on ℋ s.t. for each v∈ X, the induced subgraph Q^*[ℋ_v] is connected, where ℋ_v={H∈ℋ: v∈ H}. We present sufficient conditions on the host graph and hyperedges so that the resulting support comes from a restricted family. We primarily study two classes of graphs: (1) If the host graph has genus g and the hypergraphs satisfy a topological condition of being cross-free, then there is a primal and a dual support of genus at most g. (2) If the host graph has treewidth t and the hyperedges satisfy a combinatorial condition of being non-piercing, then there exist primal and dual supports of treewidth O(2^t). We show that this exponential blow-up is sometimes necessary. As an intermediate case, we also study the case when the host graph is outerplanar. Finally, we show applications of our results to packing and covering, and coloring problems on geometric hypergraphs.


page 1

page 2

page 3

page 4


Treewidth versus clique number in graph classes with a forbidden structure

Treewidth is an important graph invariant, relevant for both structural ...

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Let v(F) denote the number of vertices in a fixed connected pattern grap...

Subexponential-time algorithms for finding large induced sparse subgraphs

Let C and D be hereditary graph classes. Consider the following problem:...

Structural domination and coloring of some (P_7, C_7)-free graphs

We show that every connected induced subgraph of a graph G is dominated ...

Short Plane Supports for Spatial Hypergraphs

A graph G=(V,E) is a support of a hypergraph H=(V,S) if every hyperedge ...

Patch Graph Rewriting

The basic principle of graph rewriting is the stepwise replacement of su...

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

In this paper we study self-similar and fractal networks from the combin...

Please sign up or login with your details

Forgot password? Click here to reset