Homomorphism Reconfiguration via Homotopy

Wrochna, Marcin

doi:10.1137/17m1122578

Cited by 23 publications

(52 citation statements)

References 33 publications

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“…This result fits the general theme, which first emerged in the work of Wrochna [21], that the complexity of H-Recolouring may be mostly determined by the structure of a topological complex in which vertices, edges and 4-cycles of H (and, more generally, complete bipartite subgraphs) are faces. That is, Wrochna's result [21] says that, if this complex is "thin" in the sense that all of its faces are 0-or 1-dimensional, then its simple topological structure can be exploited to obtain a polynomial-time algorithm, whereas our result says that if this complex has basically the same topology as a 2-sphere, then the problem is PSPACE-complete. We will further discuss the potential connections between H-Recolouring and topology at the end of the paper.…”

Section: Introductionsupporting

confidence: 86%

“…The complexity results obtained for H-Recolouring thus far seem to be pointing towards a dichotomy (P/PSPACE-complete) based, at least partially, on the structure of a topological complex defined in terms of H. For irreflexive H, the important complex seems to be a CW-complex defined in terms of H × K 2 , in which edges are 1-dimensional cells, 4-cycles are 2-dimensional cells and larger complete bipartite subgraphs are higher dimensional cells. While a very vague picture of this potential dichotomy has started to materialize from the results of Wrochna [21] and our Theorem 1.1, it seems that much more data will be required before one can make a plausible conjecture regarding the full complexity classification. A natural place to start, which is probably difficult in its own right, might be to classify the complexity in the case of irreflexive K 2,3 -free graphs H (perhaps with the additional assumption that H is bipartite).…”

Section: Resultsmentioning

confidence: 93%

“…1 Later, Brewster, McGuinness, Moore and Noel [5] extended this dichotomy to the case when H is a "circular clique. " Wrochna [21] developed ideas inspired by algebraic topology to prove the remarkably general result that H-Recolouring is solvable in polynomial time whenever H does not contain a cycle of length 4. By further refining his topological approach, Wrochna [22] (see also [23]) proved a "multiplicativity" result for graphs without cycles of length 4 which is closely connected to Hedetniemi's Conjecture [11]; very recently, Tardif and Wrochna [19] have extended these methods beyond the setting of C 4 -free graphs.…”

Section: Introductionmentioning

confidence: 99%

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Reconfiguring graph homomorphisms on the sphere

Lee

Noel

Siggers

2020

European Journal of Combinatorics

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Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle.If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which H-Recolouring is known to be PSPACE-complete for reflexive instances.

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Section: Introductionsupporting

confidence: 86%

Section: Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Reconfiguring graph homomorphisms on the sphere

Lee

Noel

Siggers

2020

European Journal of Combinatorics

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“…. , c k } holds for every vertex v. These problems have been studied intensively from various viewpoints [1-4, 7, 8, 14, 17, 20] including the generalizations [6,21]. Bonsma and Cereceda [2] proved that coloring reconfiguration is PSPACE-complete even for bipartite graphs and any fixed constant k ≥ 4.…”

Section: Known and Related Resultsmentioning

confidence: 99%

Parameterized complexity of the list coloring reconfiguration problem with graph parameters

Hatanaka

Ito

Zhou

2018

Theoretical Computer Science

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Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.Recently, the framework of reconfiguration [16] has been extensively studied in the field of theoretical computer science. This framework models several situations where we wish to find a step-by-step transformation between two feasible solutions of a combinatorial (search) problem such that all intermediate solutions are also feasible and each step respects a fixed reconfiguration rule. This reconfiguration framework has been applied to several well-studied combinatorial problems. (See a survey [15].) Our problemIn this paper, we study a reconfiguration problem for list (vertex) colorings in a graph, which was introduced by Bonsma and Cereceda [2].

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“…A particular case is that of n = 3; X d C 3 is the space of proper 3-colourings of Z d . This condition was studied in [23] in the context of reconfiguration problems; we remark that the so-called fundamental groupoid in that paper is intimately related to the universal cover of H. If H = C 3 then the lifts correspond to the so called height functions ( [14]).…”

Section: Phased Mixing Properties For Four-cycle Hom-free Graphsmentioning

confidence: 99%

Mixing properties for hom-shifts and the distance between walks on associated graphs

Chandgotia

Marcus

2018

Pacific J. Math.

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Let H be a finite connected undirected graph and H 2 walk be the graph of bi-infinite walks on H; two such walks {xi} i∈Z and {yi} i∈Z are said to be adjacent if xi is adjacent to yi for all i ∈ Z. We consider the question: Given a graph H when is the diameter (with respect to the graph metric) of H 2 walk finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of Z d with respect to the standard generators to H) and are the subject of this paper.

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Homomorphism Reconfiguration via Homotopy

Cited by 23 publications

References 33 publications

Reconfiguring graph homomorphisms on the sphere

Reconfiguring graph homomorphisms on the sphere

Parameterized complexity of the list coloring reconfiguration problem with graph parameters

Mixing properties for hom-shifts and the distance between walks on associated graphs

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