Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices {s_i,t_i} of G, find a minimum set of edges of G whose removal disconnects each s_i from the corresponding t_i. Multicut is known to be NP-hard and Max SNP-hard even when the input graph is restricted to being a tree. The main result of the paper is a polynomial-time approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any ε>0, we present a polynomial-time (1+ε)-approximation algorithm. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provide some hardness results: we prove that Multicut is still NP-hard for binary trees and that it is Max SNP-hard if we drop any of the three conditions (unweighted, bounded-degree, bounded tree-width). Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut.     

On the Colin de Verdière number of graphs

Let μ(G) and ω(G) be the Colin de Verdière and clique numbers of a graph G, respectively. It is well-known that μ(G)?ω(G)-1 for all graphs. Our main results include μ(G)?ω(G) for all chordal graphs; μ(G)?tw(G)+1 for all graphs (where tw is the tree-width), and a characterization of those split (⊆ chordal) graphs for which μ(G)=ω(G). The bound μ(G)?tw(G)+1 improves a result of Colin de Verdière by a factor of 2.     

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C₁,…,C_k} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.     

Parameterized complexity of spare capacity allocation and the multicost Steiner subgraph problem

We study the computational complexity of the Spare Capacity Allocation problem arising in optical networks that use a shared mesh restoration scheme. In this problem we are given a network with edge capacities and point-to-point demands, and the goal is to allocate two edge-disjoint paths for each demand (a working path and a so-called restoration path, which is activated only if the working path fails) so that the capacity constraints are satisfied and the total cost of the used and reserved bandwidth is minimized. We focus on the setting where we deal with a group of demands together, and select their restoration paths simultaneously in order to minimize the total cost. We investigate how the computational complexity of this problem is affected by certain parameters, such as the number of restoration paths to be selected, or the treewidth of the network graph. To analyze the complexity of the problem, we introduce a generalization of the Steiner Forest problem that we call Multicost Steiner Subgraph. We study its parameterized complexity, and identify computationally easy and hard cases by providing hardness proofs as well as efficient (fixed-parameter tractable) algorithms.     

Line graphs of bounded clique-width

We show that a set of graphs has bounded tree-width or bounded path-width if and only if the corresponding set of line graphs has bounded clique-width or bounded linear clique-width, respectively. This relationship implies some interesting algorithmic properties and re-proves already known results in a very simple way. It also shows that the minimization problem for NLC-width is NP-complete.     

On the OBDD size for graphs of bounded tree- and clique-width

We study the size of OBDDs (ordered binary decision diagrams) for representing the adjacency function f_G of a graph G on n vertices. Our results are as follows:

-

for graphs of bounded tree-width there is an OBDD of size O(logn) for f_G that uses encodings of size O(logn) for the vertices;

-

for graphs of bounded clique-width there is an OBDD of size O(n) for f_G that uses encodings of size O(n) for the vertices;

-

for graphs of bounded clique-width such that there is a clique-width expression for G whose associated binary tree is of depth O(logn) there is an OBDD of size O(n) for f_G that uses encodings of size O(logn) for the vertices;

-

for cographs, i.e. graphs of clique-width at most 2, there is an OBDD of size O(n) for f_G that uses encodings of size O(logn) for the vertices. This last result complements a recent result by Nunkesser and Woelfel [R. Nunkesser, P. Woelfel, Representation of graphs by OBDDs, in: X. Deng, D. Du (Eds.), Proceedings of ISAAC 2005, in: Lecture Notes in Computer Science, vol. 3827, Springer, 2005, pp. 1132-1142] as it reduces the size of the OBDD by an O(logn) factor using encodings whose size is increased by an O(1) factor.

     

Linear connectivity forces large complete bipartite minors: An alternative approach

The recent paper ‘Linear connectivity forces large complete bipartite minors’ by Böhme, Kawarabayashi, Maharry and Mohar relies on an extension of Robertson and Seymour?s structure theorem for graphs with a forbidden minor. We describe a more direct approach which uses just the original structure theorem.