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Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges

Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, Jan Kratochvíl

TL;DR

NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones, and previously known results for covering graphs without semi-edges are strengthened.

Abstract

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em 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, Fellows, and Stillwell 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 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).

Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges

TL;DR

NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones, and previously known results for covering graphs without semi-edges are strengthened.

Abstract

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em 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, Fellows, and Stillwell 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 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 -tuples of pairwise disjoint perfect matchings in regular graphs).

Paper Structure

This paper contains 12 sections, 23 theorems, 11 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Graph coverings and complexity
  3. Graphs with semi-edges
  4. Formal definitions
  5. Overview of our results
  6. The impact of semi-edges
  7. One-vertex target graphs
  8. Two-vertex target graphs
  9. Polynomial parts of Theorem \ref{['t:2-vertex']}
  10. NP-hardness for non-regular target graphs
  11. NP-hardness for connected regular target graphs
  12. Conclusion

Key Result

Proposition 1

A graph $G$ covers a connected graph $H$ if and only if $G$ allows a pair of mappings $f_V:V(G)\longrightarrow V(H)$ and $f_\Lambda:\Lambda(G)\longrightarrow \Lambda(H)$ such that

Figures (20)

  • Figure 1: An example of a graph presented in a usual graph-theoretical way (left) and using the dart-based Definition \ref{['def:graph-dart']} (right).
  • Figure 2: An example of a covering. The vertex mapping of the covering from $G$ to $H$ is determined by the shape of the vertices, the edge mapping by the colors of the edges.
  • Figure 3: Examples of small semi-simple graphs which define NP-complete covering problems.
  • Figure 4: An illustration for the construction of the graph $G'$ in the proof of Theorem \ref{['t-onevertex-general']}.
  • Figure 5: A gadget $G_{3,4}$ from Proposition \ref{['t:2vertex-np-differentdegrees']}.
  • ...and 15 more figures

Theorems & Definitions (50)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • Proposition 1
  • definition 5
  • definition 6
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 40 more