Asymptotic behavior of spanning forests and connected spanning subgraphs on two-dimensional lattices

Two-connected Spanning Subgraphs with at Most

Packing of 2-Connected Spanning Subgraphs and Spanning Trees

We prove that every (6k + 2`, 2k)-connected simple graph contains k rigid and ` connected edgedisjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach p...

On finding highly connected spanning subgraphs

In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph G and an integer ruv for every pair of vertices u, v ∈ V (G). The objective is to construct a subgraph H of minimum weight which contains ruv edge-disjoint (or node-disjoint) u-v paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory a...

Generating k-Vertex Connected Spanning Subgraphs and k-Edge Connected Spanning Subgraphs

We show that k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k. We also show that generating k-edge connected spanning subgraphs, where k is part of the input, can be done in incremental polynomial time. These results are based on properties of minimally k-connected graphs which might be of independent interest.

Spanning Forests on Random Planar Lattices

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q → 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n → −1, the expansion parameter t counting the number of c...