Network fault costs based on minimum leaf spanning trees

TL;DR

The paper investigates network fault-tolerance through the lens of the minimum leaf number $ml(G)$ and its behavior under vertex deletion, introducing leaf-guaranteed graphs and the fault cost $\varphi(G)$. It develops both structural theory and a computational framework to analyze $\varphi(G)$, including an algorithm that enumerates ml-subgraphs and computes transition costs between the fault-free and vertex-deleted networks. Key contributions include a characterization of graphs with $\varphi(G)=0$, explicit constructions proving that every $k\ge0$ occurs as a fault cost, and detailed treatment of the challenging $k=1$ case along with infinite families for $k=3$ in cubic graphs. The work advances understanding of fault-tolerant network design by linking combinatorial properties of spanning trees to practical fault-cost metrics and raises open questions about higher connectivity and specific graph classes. These results have potential implications for designing robust networks with predictable minimal adjustment costs after failures.

Abstract

We study the fault-tolerance of networks from both the structural and computational point of view using the minimum leaf number of the corresponding graph $G$, i.e. the minimum number of leaves of the spanning trees of $G$, and its vertex-deleted subgraphs. We investigate networks that are leaf-guaranteed, i.e. which satisfy a certain stability condition with respect to minimum leaf numbers and vertex-deletion. Next to this, our main notion is the so-called fault cost, which is based on the number of vertices that have different degrees in minimum leaf spanning trees of the network and its vertex-deleted subgraphs. We characterise networks with vanishing fault cost via leaf-guaranteed graphs and describe, for any given network $N$, leaf-guaranteed networks containing $N$. We determine for all non-negative integers $k \le 8$ except $1$ the smallest network with fault cost $k$. We also give a detailed treatment of the fault cost $1$ case, prove that there are infinitely many $3$-regular networks with fault cost $3$, and show that for any non-negative integer $k$ there exists a network with fault cost exactly $k$.

Figures (12)

• Figure 1: Replacing an edge with a copy of the graph $\Xi_8$.
• Figure 2: The smallest bipartite leaf-guaranteed graph.
• Figure 3: A 2-connected graph $G$ with maximum degree and minimum leaf number $k := \deg(v) \ge 3$. In $G - v$, any spanning tree has at least $2k$ leaves. An ml-subgraph of $G - v$ with exactly $2k$ leaves is emphasised.
• Figure 4: Left-hand side (a): The graph $G_m$, $m\geq 3$, with fault cost $2\lfloor m/2 \rfloor + 2$. Right-hand side (b): The graph $H_m$, $m\geq 5$ with fault cost $2\lfloor m/2 \rfloor - 1$.
• Figure 5: Subgraphs used in the proof of Corollary \ref{['cor:family_cubic_fc3']}. Left-hand side (a): A Type 1 graph. Right-hand side (b): A Type 2 graph also satisfying the extra condition of Theorem \ref{['thm:construction_fc3']}.

Theorems & Definitions (10)

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