An exact-arithmetic algorithm for spanning tree modulus

Abstract

Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical decompositions of arbitrary graphs, one based on strength and the other on arboricity. Here we introduce an exact-arithmetic algorithm for spanning tree modulus and the strength-based decomposition using Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between spanning tree modulus and critical edge sets from the vulnerability problem. This paper introduces the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests.

Figures (5)

• Figure 1: An example of the minimum cut problem described in Section \ref{['sec:min-cut']}. The original graph $G$ is shown on the left. The goal is to increase the value of $x$ on the highlighted edge $j=\{f,a\}$ as much as possible without violating any constraints of the polymatroid defined in \ref{['eq:Pf']}. The right-hand side shows the corresponding flow graph $G'$. The maximum increment $\epsilon_{\max}(j)$ is computed from the value of the minimum cut; a tight set $J'(j)$ is obtained from the minimum cut edges.
• Figure 2: An example, described in Section \ref{['sec:obtain-crit-set']}, of how Cunningham's algorithm may not produce a critical set of edges.
• Figure 3: Steps of the spanning tree modulus applied to Zachary's karate club graph.
• Figure 4: Run-time complexity scaling tests as described in Section \ref{['sec:timing']}.
• Figure 5: Visualization of the spanning tree modulus for the C. elegans metabolic network as described in Section \ref{['sec:celegans']}.

Theorems & Definitions (31)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5