23,118 research outputs found

    Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups

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    We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in Journal of Graph Theor

    Branch-depth: Generalizing tree-depth of graphs

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    We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E)G = (V,E) and a subset AA of EE we let Ξ»G(A)\lambda_G (A) be the number of vertices incident with an edge in AA and an edge in Eβˆ–AE \setminus A. For a subset XX of VV, let ρG(X)\rho_G(X) be the rank of the adjacency matrix between XX and Vβˆ–XV \setminus X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions Ξ»G\lambda_G has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG\rho_G has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

    Strong homotopy properads

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    In this paper we prove that the structure of strong homotopy properad transfers over left homotopy inverses and give explicit formulae for the induced structure.Comment: Final version, 19 page

    Branch-depth: Generalizing tree-depth of graphs

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    We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E)G = (V,E) and a subset AA of EE we let Ξ»G(A)\lambda_G (A) be the number of vertices incident with an edge in AA and an edge in Eβˆ–AE \setminus A. For a subset XX of VV, let ρG(X)\rho_G(X) be the rank of the adjacency matrix between XX and Vβˆ–XV \setminus X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions Ξ»G\lambda_G has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG\rho_G has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio
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