A Maximum Resonant Set of Polyomino Graphs

A polyomino graph $H$ is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. In this paper, we show that if $K$ is a maximum resonant set of $H$, then $H-K$ has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to its Clar number. Based on this result, we have that the maximum forcing number of a polyomino graph can be computed in polynomial time. We also show that if $K$ is a maximal alternating set of $H$, then $H-K$ has a unique perfect matching.Comment: 13 pages, 6 figure

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Multiple object tracking with context awareness

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