
Vertex partitions of (C_3,C_4,C_6)free planar graphs
A graph is (k_1,k_2)colorable if its vertex set can be partitioned into...
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Graph Homomorphism Reconfiguration and Frozen HColourings
For a fixed graph H, the reconfiguration problem for Hcolourings (i.e. ...
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Circumscribing Polygons and Polygonizations for Disjoint Line Segments
Given a planar straightline graph G=(V,E) in R^2, a circumscribing poly...
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On the dichromatic number of surfaces
In this paper, we give bounds on the dichromatic number χ⃗(Σ) of a surfa...
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Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast
The NPhard Maximum Planar Subgraph problem asks for a planar subgraph H...
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Angle Covers: Algorithms and Complexity
Consider a graph with a rotation system, namely, for every vertex, a cir...
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Retracting Graphs to Cycles
We initiate the algorithmic study of retracting a graph into a cycle in ...
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Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertexmapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edgesigns. Special homomorphisms of signed graphs, called shomomorphisms, have been studied. In an shomomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. This concept has been extensively studied, and a full complexity classification (polynomial or NPcomplete) for shomomorphism to a fixed target signed graph has recently been obtained. Such a dichotomy is not known when we restrict the input graph to be planar (not even for nonsigned graph homomorphisms). We show that deciding whether a (nonsigned) planar graph admits a homomorphism to the square C_t^2 of a cycle with t> 6, or to the circular clique K_4t/(2t1) with t>2, are NPcomplete problems. We use these results to show that deciding whether a planar signed graph admits an shomomorphism to an unbalanced even cycle is NPcomplete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar shomomorphism problem with any signed cycle as a target. We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar shomomorphism problems to signed cycles remain NPcomplete even for inputs of maximum degree 3 (except for the case of unbalanced 4cycles, for which we show this for maximum degree 4). We also show that for a given integer g, the problem for signed bipartite planar inputs of girth g is either trivial or NPcomplete.
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