[Paper Review] On the Medianwidth of Graphs

Konstantinos S. Stavropoulos|arXiv (Cornell University)|Dec 3, 2015

Advanced Graph Theory Research41 references3 citations

TL;DR

This paper introduces median decompositions, a generalization of tree decompositions that use median graphs as the underlying structure instead of trees. It defines the $i$-medianwidth $\operatorname{mw}_i(G)$ as a hierarchy of width parameters that start at treewidth ($i=1$) and converge to the clique number $\omega(G)$ as $i$ increases, with a variant matching the chromatic number $\chi(G)$, offering a new structural framework for graph classes beyond bounded treewidth.

ABSTRACT

A median graph is a connected graph, such that for any three vertices $u,v,w$ there is exactly one vertex $x$ that lies simultaneously on a shortest $(u,v)$-path, a shortest $(v,w)$-path and a shortest $(w,u)$-path. Examples of median graphs are trees and hypercubes. We introduce and study a generalisation of tree decompositions, to be called median decompositions, where instead of decomposing a graph $G$ in a treelike fashion, we use general median graphs as the underlying graph of the decomposition. We show that the corresponding width parameter $ ext{mw}(G)$, the medianwidth of $G$, is equal to the clique number of the graph, while a suitable variation of it is equal to the chromatic number of $G$. We study in detail the $i$-medianwidth $ ext{mw}_i(G)$ of a graph, for which we restrict the underlying median graph of a decomposition to be isometrically embeddable to the Cartesian product of $i$ trees. For $i\geq 1$, the parameters $ ext{mw}_i$ constitute a hierarchy starting from treewidth and converging to the clique number. We characterize the $i$-medianwidth of a graph to be, roughly said, the largest "intersection" of the best choice of $i$ many tree decompositions of the graph. Lastly, we extend the concept of tree and median decompositions and propose a general framework of how to decompose a graph $G$ in any fixed graphlike fashion.

Motivation & Objective

• To generalize tree decompositions by replacing trees with median graphs as the underlying decomposition structure.
• To define and study the medianwidth $\operatorname{mw}(G)$, showing it equals the clique number $\omega(G)$.
• To introduce $i$-medianwidth $\operatorname{mw}_i(G)$, which restricts decompositions to median graphs isometrically embeddable in the product of $i$ trees.
• To establish that $\operatorname{mw}_i(G)$ forms a non-increasing hierarchy from treewidth to clique number.
• To explore connections between $i$-medianwidth and graph coloring, including a variant matching the chromatic number $\chi(G)$.

Proposed method

• Define median decompositions using convex subgraphs of median graphs as bags, generalizing tree decomposition concepts.
• Use the Helly property and convexity in median graphs to ensure consistent intersection behavior across bags.
• Characterize $\operatorname{mw}(G)$ as equal to the clique number $\omega(G)$ via structural properties of median graphs.
• Introduce $i$-medianwidth by restricting the underlying median graph to be isometrically embeddable in the Cartesian product of $i$ trees.
• Define weakly-$\Theta$-smooth median decompositions to relate the width parameter to the chromatic number $\chi(G)$.
• Prove that the smooth medianwidth (a variant of $\operatorname{mw}_i$) equals $\chi(G)$, establishing a duality similar to treewidth and brambles.

Experimental results

Research questions

• RQ1 How can tree decompositions be generalized using median graphs instead of trees as the underlying structure?
• RQ2 What is the relationship between the medianwidth $\operatorname{mw}(G)$ and classical graph parameters like the clique number $\omega(G)$?
• RQ3 Does the $i$-medianwidth $\operatorname{mw}_i(G)$ form a proper hierarchy converging from treewidth to the clique number?
• RQ4 Can a variant of median decomposition capture the chromatic number $\chi(G)$, and if so, how?
• RQ5 What are the algorithmic implications of $i$-medianwidth for NP-hard problems on graphs of bounded $i$-medianwidth?

Key findings

• The medianwidth $\operatorname{mw}(G)$ of a graph $G$ is exactly equal to its clique number $\omega(G)$.
• The $i$-medianwidth $\operatorname{mw}_i(G)$ forms a non-increasing hierarchy: $\operatorname{tw}(G)+1 = \operatorname{mw}_1(G) \geq \operatorname{mw}_2(G) \geq \cdots \geq \operatorname{mw}(G) = \omega(G)$.
• For $i \geq 1$, $\operatorname{mw}_i(G)$ can be interpreted as the largest 'intersection' of the best choice of $i$ tree decompositions of $G$.
• A variant of median decomposition, called weakly-$\Theta$-smooth, yields a width parameter equal to the chromatic number $\chi(G)$.
• The $i$-medianwidth hierarchy is proper: each level is unbounded in the sense that complete $i$-partite graphs achieve arbitrarily large $\operatorname{mw}_i(G)$.
• It remains unknown whether the $i$-medianwidth of a graph is fixed-parameter tractable to compute for $i \geq 2$, despite the problem being W[1]-complete for the clique number.