Smooth digraphs modulo primitive positive constructability and cyclic loop conditions

Finite smooth digraphs, that is, finite directed graphs without sources and sinks, can be partially ordered via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two smooth digraphs in our poset it suffices to show that the polymorphism clone of one of the digraphs satisfies a prime cyclic loop condition that is not satisfied by the polymorphism clone of the other. Furthermore, we prove that the poset of cyclic loop conditions ordered by their strength for clones is a distributive lattice, too.

Sharp Inequalities for the Hardy–Littlewood Maximal Operator on Finite Directed Graphs

In this paper, we introduce and study the Hardy–Littlewood maximal operator MG→ on a finite directed graph G→. We obtain some optimal constants for the ℓp norm of MG→ by introducing two classes of directed graphs.

Graph Pattern Matching: Capturing Bisimilar Subgraph

Graph pattern matching that aims to seek out answer graphs in a data graph matching a provided graph, plays a fundamental role as a part of graph search for graph databases. “Matching” indicates that the two graphs are correlated, such as bisimulation, isomorphism, simulation, etc. The strictness of bisimulation is between simulation and isomorphism. Seldom work has been done to search for bisimulation subgraphs. This research focuses on the problem. The symbol [Formula: see text] is introduced to fundamental modal logic language, thereby yielding [Formula: see text] language; the symbols [Formula: see text] is added for forming [Formula: see text] formulas. Then conclusions about graph bisimulations are shown. Subsequently, a theorem with detailed proof is presented, stating that [Formula: see text] formulas characterize finite directed graphs modulo bisimulation. According to the conclusions and theorem, algorithms for finding subgraphs are proposed. After dividing the query graph, the match graphs undergo the characterization using [Formula: see text] formulas. In the data graphs, by model checking the formulas, the answer graphs exhibiting bisimilarity to the match graphs are able to be captured.