[Paper Review] The tree of decomposition of a biconnected graph

D. V. Karpov|arXiv (Cornell University)|May 28, 2014

Advanced Graph Theory Research7 references3 citations

TL;DR

This paper introduces a novel tree of decomposition for biconnected graphs using parts of k-vertex cutset decompositions, generalizing Tutte's structure tree for 2-vertex cutsets. The method constructs a tree where nodes represent parts of the decomposition, enabling precise bounds on chromatic and choice numbers, and fully characterizes critical biconnected graphs as those with no internal vertices in any part-block or part-triangle.

ABSTRACT

The tree of decomposition of a $k$-connected graph by a set $\mathfrak S$ of pairwise independent $k$-vertex cutsets is defined as follows. The vertices of this tree are cutsets of $\mathfrak S$ and parts of decomposition of the graph by the set $\mathfrak S$, each cutset is adjacent to all parts that contain it. We prove, that the graph described above is a tree. The tree of decomposition of a biconnected graph is a particular case of this construction: it is the tree of decomposition of a biconnected graph by the set of all its single cutsets (i.e. 2-vertex cutsets, that are independent with all other 2-vertex cutsets). We show that this tree has much in common with the classic tree of blocks and cutpoints of a connected graph. With the help of the tree of decomposition of a biconnected graph we prove a planarity criterium and find some upper bounds on the chromatic number of this graph. Finally, we study the structure of critical biconnected graphs and prove that each such graph has at least four vertices of degree 2.

Motivation & Objective

• To develop a new tree-based decomposition structure for biconnected graphs that generalizes the classic block-cutpoint tree to 2-vertex cutsets.
• To provide a framework for analyzing the structural disposition of pairwise independent 2-vertex cutsets in biconnected graphs.
• To derive tight bounds on the chromatic number and choice number of biconnected graphs using the decomposition tree.
• To characterize critical biconnected graphs by identifying structural conditions under which no vertex can be removed without losing biconnectivity.
• To establish that critical biconnected graphs must have all parts-blocks and parts-triangles with empty interiors, and at least four degree-2 vertices.

Proposed method

• The paper defines a decomposition of a biconnected graph G using the set of all 2-vertex cutsets, denoted R₂(G), and constructs parts A ∈ Part(R₂(G)) such that no cutset splits A, but every vertex outside A is separated by some cutset.
• Each part A is partitioned into its interior Int(A) (vertices not in any cutset) and boundary Bound(A) (vertices in cutsets), with Int(A) being fully adjacent only to vertices within A.
• A tree BT(G) is built with nodes representing parts and cutsets, where edges connect parts to the cutsets they contain, forming a tree structure that reflects the hierarchical decomposition.
• The chromatic number χ(G) is bounded using the maximum chromatic number of any part G(A) plus a small additive term, with χ(G) ≤ max_A χ(G(A)) + 1.
• The choice number ch(G) is similarly bounded: ch(G) ≤ max_A ch(G(A)) + 2, with a refined bound when parts are cycles.
• The structure is applied to characterize critical biconnected graphs as those where all part-blocks and part-triangles have empty interiors, and terminal parts are cycles of length at least 4 with only two non-degree-2 vertices.

Experimental results

Research questions

• RQ1 How can the relative disposition of 2-vertex cutsets in a biconnected graph be systematically described using a tree-like structure?
• RQ2 What are the tightest possible bounds on the chromatic number of a biconnected graph based on its decomposition into parts separated by 2-vertex cutsets?
• RQ3 Can the choice number of a biconnected graph be bounded using the choice numbers of its decomposed parts and the structure of the decomposition tree?
• RQ4 What structural conditions define a critical biconnected graph in terms of its decomposition into parts and cutsets?
• RQ5 What is the minimum number of degree-2 vertices in a critical biconnected graph, and how is this number related to the decomposition tree?

Key findings

• The chromatic number of a biconnected graph G satisfies χ(G) ≤ max_{A∈Part(G)} χ(G(A)) + 1, with equality possible only if the maximum is achieved at a part with non-empty interior.
• The choice number of G satisfies ch(G) ≤ max_{A∈Part(G)} ch(G(A)) + 2, and this bound is tight; for cycle parts, the additive term can be reduced to 1.
• If all parts of a biconnected graph are cycles, then χ(G) ≤ 3, which provides a sufficient condition for 3-colorability.
• A biconnected graph G is critical if and only if all its part-blocks and part-triangles have empty interiors, meaning no vertex is removable without breaking biconnectivity.
• Every critical biconnected graph has at least four vertices of degree 2, and this bound is tight, as exemplified by the 4-cycle and certain graphs with a single 2-vertex cutset.
• Terminal parts in the decomposition tree of a critical biconnected graph are cycles of length at least 4, with only two vertices in the cutset and all others having degree 2 in G.