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Computer Science > Data Structures and Algorithms

arXiv:2204.09113 (cs)
[Submitted on 19 Apr 2022 (v1), last revised 17 Oct 2022 (this version, v3)]

Title:Representation of short distances in structurally sparse graphs

Authors:Zdeněk Dvořák
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Abstract:A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.
Comments: 33 pages, no figures; v3: Added an improved approximation subject to VC-dimension constraints and an application for the approximation of distance domination and independence number
Subjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
MSC classes: 05C12
Cite as: arXiv:2204.09113 [cs.DS]
  (or arXiv:2204.09113v3 [cs.DS] for this version)
  https://doi.org/10.48550/arXiv.2204.09113

Submission history

From: Zdeněk Dvořák [view email]
[v1] Tue, 19 Apr 2022 19:57:32 UTC (16 KB)
[v2] Thu, 28 Jul 2022 14:27:16 UTC (23 KB)
[v3] Mon, 17 Oct 2022 12:12:38 UTC (23 KB)
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