Adjacency list

About: Adjacency list is a research topic. Over the lifetime, 4419 publications have been published within this topic receiving 78449 citations.

Adjacency constraints in harvest scheduling: an aggregation heuristic

Abstract: An heuristic for adjacency constraint aggregation is proposed. The heuristic is composed of two procedures. Procedure 1 consists of identifying harvesting areas for which it is not necessary to wri...

Berge trigraphs

Abstract: A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one of a few kinds of decompositions. We used three kinds of decompositions: skew-partitions, 2-joins, and proper homogeneous pairs. At that time we were not sure whether all three decompositions were necessary. In this article we show that the proper homogeneous pair decomposition is in fact unnecessary. This is a consequence of a general decomposition theorem for “Berge trigraphs.” A trigraph T is a generalization of a graph, where the adjacency of some vertex pairs is “undecided.” A trigraph is Berge if however we decide the undecided pairs, the resulting graph is Berge. We show that the decomposition result of [2] for Berge graphs extends (with slight modifications) to Berge trigraphs; that is for a Berge trigraph T, either T belongs to one of a few basic classes or T admits one of a few decompositions. Moreover, the decompositions are such that, however, we decide the undecided pairs of T, the resulting graph admits the same decomposition. This last property is crucial for the application. The full proof of this result is over 200 pages long and was the author's PhD thesis. In this article we present the parts that differ significantly from the proof of the decomposition theorem for Berge graphs, and only in the case needed for the application. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 1–55, 2006 This research was partially conducted during the period the author served as a Clay Mathematics Institute Research Fellow.

Adjacency Labelling for Planar Graphs (and Beyond)

Abstract: We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every $n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$. These results generalize to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and $k$-planar graphs.

Contour filling in raster graphics

Abstract: The paper discusses algorithms for filling contours in raster graphics. Its major feature is the use of the line adjacency graph for the contour in order to fill correctly nonconvex and multiply connected regions, while starting from a “seed.” Because the same graph is used for a “parity check” filling algorithm, the two types of algorithms can be combined into one. This combination is useful for either finding a seed through a parity check, or for resolving ambiguities in parity on the basis of connectivity.