Finding a Sparse Connected Spanning Subgraph in a non-Uniform Failure Model

TL;DR

The paper studies Unweighted Flexible Graph Connectivity, a robust-connection generalization of Spanning Tree with safe and unsafe edges, focusing on parameterized complexity. It establishes an almost complete dichotomy: UFGC is FPT for key structural and problem-specific parameters (treewidth tw, distance to cluster graphs, number of unsafe edges, and a below-upper-bound solution size ℓ), while many natural parameters inherit hardness via reductions from Hamiltonian Cycle and related problems. Central techniques include backbone-based dynamic programming on tree decompositions, ear-decompositions for 2-edge-connected graphs, and Disjoint Subgraphs along with Matching to connect terminals. These results advance understanding of when robust connectivity can be achieved efficiently under non-uniform failure models and provide practical FPT algorithms for relevant graph classes. The work also outlines open questions about further parameterizations and kernelization limits in this robustness-rich setting.

Abstract

We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either \emph{safe} or \emph{unsafe} and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph $G = (V,E)$ in which the edge set $E$ is partitioned into a set $S$ of safe edges and a set $U$ of unsafe edges and the task is to find a set $T$ of at most $k$ edges such that $T - \{u\}$ is connected and spans $V$ for any unsafe edge $u \in T$. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity} admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.

Figures (1)

• Figure 1: The relations between structural graph parameters and our respective results for UFGC. A parameter $k$ is marked green () if UFGC admits an FPT-time algorithm for $k$, yellow () if it is W[1]-hard with respect to $k$, and red () if it is NP-hard for constant $k$ (para-NP-hard). We do not know the status for parameters with white boxes. An edge from a parameter $\alpha$ to a parameter $\beta$ below $\alpha$ means that there is a function $f$ such that $\beta \leq f(\alpha)$ in every graph. Hardness results for $\alpha$ imply the same hardness results for $\beta$ and an FPT-time algorithm for $\beta$ implies an FPT-time algorithm for $\alpha$.

Theorems & Definitions (11)

• Corollary 3
• Corollary 4
• Proposition 7
• Definition 8: West00
• Lemma 9
• Lemma 10
• Proposition 11
• Lemma 12
• Theorem 15
• Proposition 16