[Paper Review] Minimal Connectivity

Matthias Kriesell|arXiv (Cornell University)|Jan 12, 2011

Interconnection Networks and Systems62 references4 citations

TL;DR

This paper investigates minimal and critically k-connected graphs, focusing on edge deletion, vertex deletion, and edge contraction as reduction operations to constructively characterize k-connected graphs. It establishes that for k ≥ 4, contraction-critically k-connected graphs can be reduced via edge contractions, and proves Lovász’s conjecture on high-connectivity line graphs by leveraging edge-disjoint tree decompositions and structural properties of line graphs.

ABSTRACT

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.

Motivation & Objective

• To develop reduction methods for k-connected graphs using elementary operations like edge deletion, vertex deletion, and edge contraction.
• To understand the distribution and structural properties of reducible elements (e.g., contractible or deletable edges) in k-connected graphs.
• To investigate whether higher-connected graphs (k ≥ 5) can be reduced by contracting a bounded number of edges, addressing a key conjecture in connectivity theory.
• To verify Lovász’s conjecture on high-connectivity line graphs by analyzing connectivity in line graphs derived from edge-connected base graphs.
• To determine the existence and properties of 3-con-critically k-connected line graphs, resolving a conjecture by Slater.

Proposed method

• Uses edge deletion and contraction as primary reduction operations to analyze minimally, critically, and contraction-critically k-connected graphs.
• Applies the concept of line graphs to transform edge-connectivity problems in a base graph G into vertex-connectivity problems in L(G).
• Employs Okamura’s theorem on removable paths in edge-connected graphs to construct edge-disjoint trees covering high-degree vertices.
• Uses clique-based decomposition in line graphs: for each vertex x in G, the edges incident to x form a clique K_x in L(G), and subcliques K_x^j are defined based on edge-disjoint trees.
• Constructs a spanning subgraph H^1 in L(G) such that L(G) − E(H^1) contains k openly disjoint paths between any two vertices, ensuring k-connectivity.
• Leverages the fact that if L(G) has minimum degree ≥ 12k+11, then the set A of vertices of degree ≥ 6k+6 in G is (6k+6)-connected and G−A is edgeless, enabling tree decomposition.

Experimental results

Research questions

• RQ1 Can every 5-connected graph be reduced to a smaller 5-connected graph by contracting at most h edges, for some fixed h?
• RQ2 Does every (12k+11)-connected line graph admit a spanning tree T such that G−E(T) remains k-connected?
• RQ3 Are there any 3-con-critically k-connected line graphs, and what does this imply for Slater’s conjecture?
• RQ4 To what extent can the structure of contraction-critically k-connected graphs be characterized via line graphs of cubic graphs for k=4?
• RQ5 What is the role of edge-disjoint tree decompositions in preserving connectivity after edge removal in line graphs?

Key findings

• For k ≥ 4, contraction-critically k-connected graphs are rich and can be reduced by contracting two edges to obtain a smaller k-connected graph, suggesting a potential reduction framework for k=5.
• Conjecture 1.3 is supported: every 5-connected graph on at least b vertices can be reduced by contracting at most h edges, with b and h independent of the graph size.
• The paper proves that every (12k+11)-connected line graph has a spanning tree T such that G−E(T) is k-connected, confirming Conjecture 7.8 for line graphs.
• It is shown that if L(G) is (6k+6)-connected and has minimum degree ≥ 12k+11, then G has a set A of vertices with degree ≥ 6k+6 that is (6k+6)-connected and G−A is edgeless.
• The existence of 2k+2 edge-disjoint trees covering A in G enables the construction of k+1 edge-disjoint subgraphs T_1,…,T_{k+1} such that each covers A and has minimum degree ≥2 in each T_j.
• The paper confirms that there is no 3-con-critically k-connected line graph, thereby resolving Slater’s conjecture in the negative for this class.