[Paper Review] The minimum stretch spanning tree problem for typical graphs

Lan Lin, Yixun Lin|arXiv (Cornell University)|Dec 10, 2017

Advanced Graph Theory Research19 references3 citations

TL;DR

This paper solves the minimum stretch spanning tree problem for several classic graph families by deriving exact values of the tree-stretch invariant 𝜎𝑇(G). It establishes tight lower bounds using face level analysis in planar graphs and constructs optimal spanning trees that achieve these bounds, providing exact results for Petersen graphs, complete k-partite graphs, split graphs, generalized convex graphs, and various planar grids including rectangular, triangular, and triangulated-rectangular grids.

ABSTRACT

With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved to be NP-hard and fixed-parameter polynomial algorithms have been obtained for some special classes of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact optimality representations for Petersen graph, the complete k-partite graphs, split graphs, generalized convex graphs, and several planar grids, including rectangular grids, triangular grids, and triangular-rectangular grids.

Motivation & Objective

• To determine the exact value of the tree-stretch invariant 𝜎𝑇(G) for fundamental graph classes.
• To characterize the minimum stretch spanning tree problem in terms of fundamental cycles and face levels in planar graphs.
• To establish tight lower bounds for the stretch using geometric and combinatorial properties of planar embeddings.
• To construct optimal spanning trees that achieve the theoretical lower bounds across multiple graph families.
• To extend known results on tree-congestion to the dual problem of tree-stretch, providing a parallel theory for spanning tree optimization.

Proposed method

• Uses face level analysis in planar graphs to derive a lower bound: 𝜎𝑇(G) ≥ 𝜆max + 1, where 𝜆max is the maximum face level.
• Applies dual graph (plane dual) techniques to map spanning trees to edge-cuts and fundamental cycles.
• Constructs optimal spanning trees by selecting vertical and horizontal lines in grid embeddings, with adjustments for parity in triangulated grids.
• Employs induction on grid dimensions to prove the maximum face level 𝜆max(G) = m−1 for triangulated-rectangular grids Tm,n.
• Validates optimality by showing constructed trees achieve the lower bound via path and cycle length analysis.
• Uses shortest path and boundary cycle analysis in the dual graph to identify fundamental cycles of minimal maximum length.

Experimental results

Research questions

• RQ1 What is the exact value of the tree-stretch 𝜎𝑇(G) for the Petersen graph?
• RQ2 How does the tree-stretch behave in complete k-partite graphs and split graphs?
• RQ3 What is the minimum stretch achievable in generalized convex graphs and planar grids?
• RQ4 Can tight lower bounds on 𝜎𝑇(G) be derived using face level and dual graph properties?
• RQ5 What spanning tree structures achieve the minimum stretch in rectangular, triangular, and triangulated-rectangular grids?

Key findings

• For the Petersen graph, the minimum stretch spanning tree has 𝜎𝑇(G) = 3.
• For complete k-partite graphs Kn₁,n₂,…,nk, the exact tree-stretch is determined as 𝜎𝑇(G) = 2 when k ≥ 3 and all ni ≥ 2.
• For split graphs, the paper establishes that 𝜎𝑇(G) = 2 if the graph is not complete, and 𝜎𝑇(G) = 1 if it is complete.
• For rectangular grids Pm × Pn with m ≤ n, the minimum stretch is 𝜎𝑇(G) = m.
• For triangular grids Tn, the minimum stretch is 𝜎𝑇(G) = n.
• For triangulated-rectangular grids Tm,n with 2 ≤ m ≤ n, the minimum stretch is 𝜎𝑇(G) = m, achieved by a spanning tree constructed from columns and diagonal edges.