Generalization of matching extensions in graphs (II)

Proposed as a general framework, Liu and Yu(Discrete Math. 231 (2001) 311-320) introduced $(n,k,d)$-graphs to unify the concepts of deficiency of matchings, $n$-factor-criticality and $k$-extendability. Let $G$ be a graph and let $n,k$ and $d$ be non-negative integers such that $n+2k+d\leq |V(G)|-2$ and $|V(G)|-n-d$ is even. If when deleting any $n$ vertices from $G$, the remaining subgraph $H$ of $G$ contains a $k$-matching and each such $k$- matching can be extended to a defect-$d$ matching in $H$, then $G$ is called an $(n,k,d)$-graph. In \cite{Liu}, the recursive relations for distinct parameters $n, k$ and $d$ were presented and the impact of adding or deleting an edge also was discussed for the case $d = 0$. In this paper, we continue the study begun in \cite{Liu} and obtain new recursive results for $(n,k,d)$-graphs in the general case $d \geq0$.Comment: 12 page

Relative Expressive Power of Navigational Querying on Graphs

Motivated by both established and new applications, we study navigational query languages for graphs (binary relations). The simplest language has only the two operators union and composition, together with the identity relation. We make more powerful languages by adding any of the following operators: intersection; set difference; projection; coprojection; converse; and the diversity relation. All these operators map binary relations to binary relations. We compare the expressive power of all resulting languages. We do this not only for general path queries (queries where the result may be any binary relation) but also for boolean or yes/no queries (expressed by the nonemptiness of an expression). For both cases, we present the complete Hasse diagram of relative expressiveness. In particular the Hasse diagram for boolean queries contains some nontrivial separations and a few surprising collapses.Comment: An extended abstract announcing the results of this paper was presented at the 14th International Conference on Database Theory, Uppsala, Sweden, March 201

Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online database, and we give three short examples of the use of this database.Comment: 22 page