Distance-preserving Subgraphs of Interval Graphs
Kshitij Gajjar, Jaikumar Radhakrishnan

TL;DR
This paper investigates the complexity and bounds of constructing small, distance-preserving subgraphs in interval graphs, providing tight bounds and demonstrating the differences between branching vertices and edges.
Contribution
It introduces tight bounds for distance-preserving subgraphs in interval graphs and highlights the separation between branching vertices and edges.
Findings
Optimal distance-preserving subgraphs are NP-hard to find.
Every interval graph has an O(k) branching vertex subgraph with +1 approximation.
Lower bound of Ω(k log k) branching vertices for some interval graphs.
Abstract
We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is -hard for general graphs. Then, we show that every interval graph admits a subgraph with branching vertices that approximates pairwise terminal distances up to an additive term of . We also present an interval graph for which the approximation is necessary to obtain the upper bound on the number of branching vertices. In particular, any distance-preserving subgraph of has branching vertices. Furthermore, we prove that every interval graph admits a distance-preserving subgraph with branching vertices, implying that the lower bound for interval…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Limits and Structures in Graph Theory
