54 research outputs found

    On retracts, absolute retracts, and folds in cographs

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    Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

    Beyond Helly graphs: the diameter problem on absolute retracts

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    Characterizing the graph classes such that, on nn-vertex mm-edge graphs in the class, we can compute the diameter faster than in O(nm){\cal O}(nm) time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph HH of a graph GG is called a retract of GG if it is the image of some idempotent endomorphism of GG. Two necessary conditions for HH being a retract of GG is to have HH is an isometric and isochromatic subgraph of GG. We say that HH is an absolute retract of some graph class C{\cal C} if it is a retract of any G∈CG \in {\cal C} of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized O~(mn)\tilde{\cal O}(m\sqrt{n}) time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of kk-chromatic graphs, for every fixed k≥3k \geq 3. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively

    LSLS-Category of Moment-Angle Manifolds, Massey Products, and a Generalization of the Golod Property

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    We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes and show how this relates to vanishing of Massey products in TorR[v1,…,vn]+(R[K],R)\mathrm{Tor}^+_{R[v_1,\ldots,v_n]}(R[K],R). In particular, we characterise the Lusternik-Schnirelmann category of moment-angle manifolds ZK\mathcal{Z}_K over triangulated dd-spheres KK for d≤2d\leq 2, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. This characterisation is given in terms of the combinatorics of KK, the cup product length of H∗(ZK)H^*(\mathcal{Z}_K), as well as a certain generalisation of the Golod property. Some applications include information about the category and vanishing of Massey products for moment-angle complexes over fullerenes and kk-neighbourly complexes.Comment: New examples adde

    Building blocks for the variety of absolute retracts

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    AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1

    Loop space decompositions of moment-angle complexes associated to graphs

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    We prove that the loop space of moment-angle complexes associated to a graph is homotopy equivalent to a finite type product of spheres and loops on simply connected spheres. To do this, a general result showing that the retract of a finite type product of spheres and loops on simply connected spheres is of the same form is proved.Comment: 21 pages, comments welcom

    The homotopy theory of polyhedral products associated with flag complexes

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    If KK is a simplicial complex on mm vertices the flagification of KK is the minimal flag complex KfK^f on the same vertex set that contains KK. Letting LL be the set of vertices, there is a sequence of simplicial inclusions L→K→KfL\to K\to K^f. This induces a sequence of maps of polyhedral products (X‾,A‾)L⟶g(X‾,A‾)K⟶f(X‾,A‾)Kf(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}. We show that Ωf\Omega f and Ωf∘Ωg\Omega f\circ\Omega g have right homotopy inverses and draw consequences. For a flag complex KK the polyhedral product of the form (CY‾,Y‾)K(\underline{CY},\underline Y)^K is a co-HH-space if and only if the 11-skeleton of KK is a chordal graph, and we deduce that the maps ff and f∘gf\circ g have right homotopy inverses in this case.Comment: 25 page

    A Brightwell-Winkler type characterisation of NU graphs

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    In 2000, Brightwell and Winkler characterised dismantlable graphs as the graphs HH for which the Hom-graph Hom(G,H){\rm Hom}(G,H), defined on the set of homomorphisms from GG to HH, is connected for all graphs GG. This shows that the reconfiguration version ReconHom(H){\rm Recon_{Hom}}(H) of the HH-colouring problem, in which one must decide for a given GG whether Hom(G,H){\rm Hom}(G,H) is connected, is trivial if and only if HH is dismantlable. We prove a similar starting point for the reconfiguration version of the HH-extension problem. Where Hom(G,H;p){\rm Hom}(G,H;p) is the subgraph of the Hom-graph Hom(G,H){\rm Hom}(G,H) induced by the HH-colourings extending the HH-precolouring pp of GG, the reconfiguration version ReconExt(H){\rm Recon_{Ext}(H)} of the HH-extension problem asks, for a given HH-precolouring pp of a graph GG, if Hom(G,H;p){\rm Hom}(G,H;p) is connected. We show that the graphs HH for which Hom(G,H;p){\rm Hom}(G,H;p) is connected for every choice of (G,p)(G,p) are exactly the NU{\rm NU} graphs. This gives a new characterisation of NU{\rm NU} graphs, a nice class of graphs that is important in the algebraic approach to the CSP{\rm CSP}-dichotomy. We further give bounds on the diameter of Hom(G,H;p){\rm Hom}(G,H;p) for NU{\rm NU} graphs HH, and show that shortest path between two vertices of Hom(G,H;p){\rm Hom}(G,H;p) can be found in parameterised polynomial time. We apply our results to the problem of shortest path reconfiguration, significantly extending recent results.Comment: 17 pages, 1 figur

    LS-category of moment-angle manifolds, Massey products, and a generalisation of the Golod property

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    This paper is obtained as as synergy of homotopy theory, commutative algebra and combinatorics. We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes and show how this relates to vanishing of Massey products in Tor+R[v1,...,vn] (R [K], R) for the Stanley-Reisner ring R[K]. In particular, we characterise the Lusternik-Schnirelmann category of moment-angle manifolds ZK over triangulated d-spheres K for d ≤ 2, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. This characterisation is given in terms of the combinatorics of K, the cup product length of H* (ZK), as well as a certain generalisation of the Golod property. As an application, we describe conditions for vanishing of Massey products in the case of fullerenes and k-neighbourly complexes
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