Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F21%3A00119288" target="_blank" >RIV/00216224:14330/21:00119288 - isvavai.cz</a>

Alternative codes found

RIV/00216208:11320/21:10436970

Result on the web

<a href="http://dx.doi.org/10.4230/LIPIcs.MFCS.2021.21" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.MFCS.2021.21</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.MFCS.2021.21" target="_blank" >10.4230/LIPIcs.MFCS.2021.21</a>

Result language

angličtina

Original language name

Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases

Original language description

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs).

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

ISBN

9783959772013

ISSN

1868-8969

e-ISSN

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Number of pages

15

Pages from-to

„21:1“-„21:15“

Publisher name

Schloss Dagstuhl -- Leibniz-Zentrum f{"u}r Informatik

Place of publication

Dagstuhl

Event location

Tallin

Event date

Aug 23, 2021

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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