
Finding a MaximumWeight Convex Set in a Chordal Graph
We consider a natural combinatorial optimization problem on chordal grap...
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Colouring NonEven Digraphs
A colouring of a digraph as defined by Erdos and NeumannLara in 1980 is...
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Computing Subset Feedback Vertex Set via Leafage
Chordal graphs are characterized as the intersection graphs of subtrees ...
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Reconfiguration of Colorable Sets in Classes of Perfect Graphs
A set of vertices in a graph is ccolorable if the subgraph induced by t...
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Odd wheels are not odddistance graphs
An odd wheel graph is a graph formed by connecting a new vertex to all v...
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Reconfiguring Graph Homomorphisms on the Sphere
Given a loopfree graph H, the reconfiguration problem for homomorphisms...
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Continuous mean distance of a weighted graph
We study the concept of the continuous mean distance of a weighted graph...
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Flip distances between graph orientations
Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider socalled αorientations of a graph G, in which every vertex v has a specified outdegree α(v), and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two αorientations of a planar graph G is at most two is complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance betwe en graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or viceversa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang, Qian, and Zhang.
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