We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph dualization problem, even in the case of acyclic convex geometries, i.e., closure systems admitting an acyclic implicational base. This paper studies this case with a focus on the degree, which corresponds to the maximal number of implications in which an element occurs. We show that the problem is tractable for bounded values of this parameter, even when relaxed to the notions of premise-and conclusion-degree. Our algorithms rely on structural properties of acyclic convex geometries and involve various techniques from algorithmic enumeration such as solution graph traversal, saturation techniques, and a sequential approach leveraging from acyclicity. They are shown to perform in incremental-polynomial time. Finally, we complete these results by showing that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.
Let Q be an r-graph on q vertices and let cx > 1. The (cx, Q)-spectral radius lambda(alpha,Q)(H) of an r-graph H is defined as the maximum value of the Q-Lagrangian polynomial over all vectors with unit cx-norm. Given a family F of r-graphs, the (cx, Q)-spectral Turan number spex(alpha)(n, Q, F) of F is the maximum (cx, Q)-spectral radius lambda(alpha,Q)(H) among all n-vertex F-free r-graphs H. This spectral parameter provides a spectral counterpart of the generalized Turan problem and contains many classical spectral extremal problems as special cases. In this paper, we first prove that for any cx > 1 and every 2-covered r-graph Q with q <= pound vertices, the complete balanced -partite pound r-graph T (R)(& ell;,n) uniquely maximizes lambda(alpha,Q)(H) among all n-vertex -partite pound r-graphs. Based on this result and spectral stability theorems, we further determine the exact value of spex(alpha)(n, Q, H (R)(F)) for all sufficiently large n, where H (R)(F )stands for the expansion of an edge-critical graph F with chromatic number l + 1. As corollaries, we obtain several Turan-type results for graphs and r-graphs, including spectral bounds for the clique spectral radius of F-free graphs and the cx-spectral radius of H (R)(F)-free r-graphs, where F is an edge-critical graph.
A graph is called a threshold graph if it is induced-{P4, C4, 2K2} free. In 2020, Qiao and Zhan [J. Graph Theory 93 (2020) 222-229] determined the minimum number of Hamilton cycles in a Hamiltonian threshold graph and characterized the extremal graphs attaining this minimum. We show that every Hamiltonian threshold graph contains a special spanning subgraph, from which their two results follow directly.
Let G be a balanced bipartite graph with a bipartite adjacency matrix A. We call G det-extremal if det(A)= per(A). If every edge of G belongs to a perfect matching, then G is matching covered. Det-extremal cubic bipartite graphs of connectivity 3 and 2 have been characterized by McCuaig and Funk et al., respectively. In this paper, we extend the above results by investigating the structure of det-extremal bipartite graphs. Due to the fact that only the edges belonging to perfect matchings contribute to the det-extremal property, we focus on det-extremal bipartite graphs that are matching covered. Firstly, we derive that a matching covered plane bipartite graph is det-extremal if and only if it has a plane bipartite ear decomposition such that each interior face is of size two modulo four. Then we turn to general matching covered bipartite graph, and we associate its bipartite ear decomposition with a standard perfect matching M*. By establishing the properties of its M*-alternating cycles, we provide the characterization of a general matching covered det-extremal bipartite graph in terms of standard bipartite ear decomposition.
In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has K5 or K2,2,2 as a minor. Mills and Turner proved an analog of this theorem by showing that every 3-connected binary matroid in which every co circuit has size at least four has F7, M & lowast;(K3,3), M(K5), or M(K2,2,2) as a minor. Generalizing these results, this paper proves that every simple matroid in which all co circuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements.
