36,293 research outputs found

    Interpretations of Association Rules by Granular Computing

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    We present interpretations for association rules. We first introduce Pawlak's method, and the corresponding algorithm of finding decision rules (a kind of association rules). We then use extended random sets to present a new algorithm of finding interesting rules. We prove that the new algorithm is faster than Pawlak's algorithm. The extended random sets are easily to include more than one criterion for determining interesting rules. We also provide two measures for dealing with uncertainties in association rules

    Heavy subgraphs, stability and hamiltonicity

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    Let GG be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that GG is 2-heavy if every induced claw (K1,3K_{1,3}) of GG contains two end-vertices each one has degree at least V(G)/2|V(G)|/2; and GG is o-heavy if every induced claw of GG contains two end-vertices with degree sum at least V(G)|V(G)| in GG. In this paper, we introduce a new concept, and say that GG is \emph{SS-c-heavy} if for a given graph SS and every induced subgraph GG' of GG isomorphic to SS and every maximal clique CC of GG', every non-trivial component of GCG'-C contains a vertex of degree at least V(G)/2|V(G)|/2 in GG. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and NN-c-heavy graph is hamiltonian, where NN is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs SS such that every 2-connected o-heavy and SS-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones Mathematicae Graph Theor

    On path-quasar Ramsey numbers

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    Let G1G_1 and G2G_2 be two given graphs. The Ramsey number R(G1,G2)R(G_1,G_2) is the least integer rr such that for every graph GG on rr vertices, either GG contains a G1G_1 or G\overline{G} contains a G2G_2. Parsons gave a recursive formula to determine the values of R(Pn,K1,m)R(P_n,K_{1,m}), where PnP_n is a path on nn vertices and K1,mK_{1,m} is a star on m+1m+1 vertices. In this note, we first give an explicit formula for the path-star Ramsey numbers. Secondly, we study the Ramsey numbers R(Pn,K1Fm)R(P_n,K_1\vee F_m), where FmF_m is a linear forest on mm vertices. We determine the exact values of R(Pn,K1Fm)R(P_n,K_1\vee F_m) for the cases mnm\leq n and m2nm\geq 2n, and for the case that FmF_m has no odd component. Moreover, we give a lower bound and an upper bound for the case n+1m2n1n+1\leq m\leq 2n-1 and FmF_m has at least one odd component.Comment: 7 page