Molecular structures and Drug-Drug Interactions (DDI) are recognized as important knowledge to guide medication recommendation (MR) tasks, medical concept embedding has been applied boost their performance. Though promising performance achieved by leveraging Graph Neural Network (GNN) models encode the molecular of medications or/and DDI, we observe that existing still defective: 1) differentia...

We show that the edges of every 3-connected planar graph except K4 can be colored with two colors so that every embedding of the graph in S is asymmetric, and we characterize all planar graphs whose edges can be 2-colored so that every embedding of the graph in S is asymmetric.

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree ∆, then G is entirely 7-choosable if ∆ ≤ 4 and G is entirely (∆+2)-choosable if ∆ ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,∆+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is pr...

We construct an embedding of any right-angled Artin group into a graph braid group. We include an observation which decreases the number of strands of the graph braid group required for this embedding, yielding an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.

The probability that an embedding of a graph in Z 3 is knotted is investigated. For any given graph (embeddable in Z 3) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to innnity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particula...

A 2-page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2-page book embedding is known to be NP-complete. In this paper, we study the problem of testing whether a given graph admits a 2...