Abstract
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semiedges. 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 socalled semiedges. Semiedges 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 semiedges 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 twovertex (multi)graphs with semiedges. Our NPhardness results are proven for simple input graphs, and in the case of regular twovertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semiedges, 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 ktuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs).
BibTeX  Entry
@InProceedings{bok_et_al:LIPIcs.MFCS.2021.21,
author = {Bok, Jan and Fiala, Ji\v{r}{\'\i} and Hlin\v{e}n\'{y}, Petr and Jedli\v{c}kov\'{a}, Nikola and Kratochv{\'\i}l, Jan},
title = {{Computational Complexity of Covering Multigraphs with SemiEdges: Small Cases}},
booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
pages = {21:121:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772013},
ISSN = {18688969},
year = {2021},
volume = {202},
editor = {Bonchi, Filippo and Puglisi, Simon J.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14461},
URN = {urn:nbn:de:0030drops144611},
doi = {10.4230/LIPIcs.MFCS.2021.21},
annote = {Keywords: graph cover, covering projection, semiedges, multigraphs, complexity}
}
Keywords: 

graph cover, covering projection, semiedges, multigraphs, complexity 
Collection: 

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

2021 
Date of publication: 

18.08.2021 