Thin trees in 8-edge-connected planar graphs

https://doi.org/10.1016/j.ipl.2018.11.009Get rights and content

Highlights

  • A graph that can be decomposed into a forest and a matching, also can be decomposed into two forests with diameter at most 12.

  • Every 8-edge-connected planar graph has two edge-disjoint 12/13-thin spanning trees.

  • For every 9-edge-connected planar graph, in polynomial time, we can find two edge-disjoint 12/13-thin spanning trees.

Abstract

Merker and Postle showed that any graph that can be decomposed into a forest and a star forest has a decomposition into two forests with diameter at most 18. Using this result, they proved any planar graph of girth at least 6 has two edge-disjoint 1819-thin spanning trees. By using a simpler version of their techniques, given a graph that can be decomposed into a forest and a matching, in polynomial time, we decompose the graph into two forests with diameter at most 12. Using this result, we are able to show that any 8-edge-connected planar graph has two edge-disjoint 1213-thin spanning trees.

Introduction

In this paper, all graphs are considered to be simple (without multiedges and without loops) unless stated otherwise.

Let G=(V,E) be a graph and let F be a subset of edges. For any SV, let δF(S) be the set of all edges of F with exactly one endpoint in S. Let ϵ be a real number with 0<ϵ<1. We say F is ϵ-thin in G if for each SV we have |δF(S)|ϵ|δG(S)|. Given a connected graph G, we are interested in constructing an ϵ-thin spanning tree in G with ϵ as small as possible. Thin spanning trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP), see [1], [2], and [3]. Goddyn [4] conjectured that every graph of sufficiently large edge-connectivity has an ϵ-thin spanning tree.

Oveis Gharan and Saberi [2] gave a polynomial-time algorithm for finding 10k-thin spanning trees in k-edge-connected planar graphs; thus, [2] requires edge-connectivity to be at least 11. Later, Merker and Postle [5] showed that any 6-edge-connected planar graph has two edge-disjoint 1819-thin spanning trees. We use simplified version of the techniques in [5] and show that any 8-edge-connected planar graph has two edge-disjoint 1213-thin spanning trees. We show this by proving Theorem 1.1 below. We define the diameter of a forest to be the maximum number among the diameters of its connected components.

Theorem 1.1

Let G=(V,E) be a connected planar graph with girth at least 8. Then, G can be decomposed into two forests with diameter at most 12.

The following lemma due to Merker and Postle [5] shows the relation between the above theorem and thin spanning trees in planar graphs. We denote the dual of a planar graph G by G.

Lemma 1.2

If G=(V,E) is a planar graph that can be decomposed into two forests F1,F2 with diameter at most d, then G has two edge-disjoint dd+1-thin spanning trees.

Notice that in the above lemma, G is a simple graph and its dual could have multiedges.

Now our result about thin trees follows easily.

Theorem 1.3

Every 8-edge-connected planar (multi)graph has two edge-disjoint 1213-thin spanning trees.

Proof

This proof is essentially the same as the proof of Corollary 3.6 in [5] and it is included for completeness.

Since G is an 8-edge-connected planar (multi)graph, its dual G=(V,E) is a simple planar graph with girth at least 8. By Theorem 1.1, each connected component Gi=(Vi,Ei) of G can be decomposed into two forests (Vi,Fi(1)) and (Vi,Fi(2)) such that each has diameter at most 12. Let H1:=(V,iFi(1)) and H2:=(V,iFi(2)). Notice that for each i=1,2, Hi is a forest with diameter at most 12. Thus, G can be decomposed into two forests with diameter at most 12. By applying Lemma 1.2 to G, we conclude that G has two edge-disjoint 1213-thin spanning trees. 

The proof of Theorem 1.1 is based on a result that was proved independently by Montassier et al. [6] and Wang and Zhang [7]. They proved that every planar graph G of girth at least 8 can be decomposed into a forest F and a matching M. In particular, if G is connected, then we can convert F to a spanning tree by adding as many matching edges as possible. Thus, we have the following result.

Theorem 1.4

Every connected planar graph of girth at least 8 can be decomposed into a spanning tree and a matching.

By a result of Borodin et al. [8], there is a polynomial-time algorithm for decomposing planar graphs of girth at least 9 into a forest and a matching. Furthermore, our proof of Theorem 1.1 gives a polynomial-time algorithm to find the two forests. Thus, for any 9-edge-connected planar graph G, in polynomial time, we can find two edge-disjoint 1213-thin spanning trees in G.

In Section 2, we construct an edge-coloring for graphs that can be decomposed into a spanning tree and a matching. In Section 3, we use this edge-coloring to prove Theorem 1.1.

Section snippets

A good edge-coloring of the graph

As we noted in the previous section, there is a correspondence between Theorem 1.1 and a particular edge-coloring of planar graphs. We define this edge-coloring below.

Definition 2.1 Good edge-coloring

Let G=(V,E) be a graph, and let c:E(G){1,2} be an edge-coloring of G. We say c is a good edge-coloring if c satisfies the following two conditions:

  • 1.

    There is no edge-monochromatic cycle.

  • 2.

    Any edge-monochromatic path has length at most 12.

From now on, by a graph G we mean a connected planar graph of girth at least 8. By Theorem 1.4, G

Proof of Theorem 1.1

In this section, we prove that c which was defined in the previous section is a good edge-coloring for G.

Notice that in both the initial coloring and the final coloring, we make sure that the colors of the M-edge and the T-edge that are oriented toward the same vertex are different. Thus, every vertex of G has indegree at most 1 in each color. This implies the following lemma due to [5].

Lemma 3.1

Let c be the edge-coloring of both G and G. Then, we have:

  • 1.

    Every monochromatic cycle in G is a dicycle in G

Acknowledgements

I thank Joseph Cheriyan for many insightful discussions. I would also like to thank Zachary Friggstad and the anonymous referee for their suggestions to enhance the presentation of the paper.

This work was supported by NSERC grant RGPIN-2014-04351 (J. Cheriyan).

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