Adjacency list

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

Total weight choosability of graphs

Abstract: Suppose the edges and the vertices of a simple graph $G$ are assigned $k$-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there any finite bound or maybe already lists of length two are sufficient? We prove that $2$-element lists are enough for trees, wheels, unicyclic and complete graphs, while the ones of length $3$ are sufficient for complete bipartite graphs. Our main tool is an algebraic theorem by Alon called Combinatorial Nullstellensatz.

Double covers of graphs

Abstract: A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

Leader-Following Consensus for High-Order Nonlinear Stochastic Multiagent Systems

Abstract: This paper considers the distributed consensus tracking problem for a class of high-order stochastic multiagent systems with uncertain nonlinear functions under a fixed undirected graph. Through the recursive method, the novel nonlinear distributed controllers are designed. By constructing a kind of special form for the virtual controller in the first step of recursive design, we realize that the state variables of every agent are separated except the outputs of the adjacency agents. The designed controller of each agent only depends on its own state variables and the outputs of the adjacent multiagents. With the proposed method, it is not required any more that the orders of the agents are same. This makes the designed controller be easier to be implemented and the proposed method be applicable for a wider class of multiagent systems. The efficiency of the design approach is illustrated by a simulation example.

A multi-resolution topological representation for non-manifold meshes

Abstract: We address the problem of representing and processing 3D objects, described through simplicial meshes, which consist of parts of mixed dimensions, and with a non-manifold topology, at different levels of detail. First, we describe a multi-resolution model, that we call a non-manifold multi-tessellation (NMT), and we consider the selective refinement query, which is at the heart of several analysis operations on multi-resolution meshes. Next, we focus on a specific instance of a NMT, generated by simplifying simplicial meshes based on vertex-pair contraction, and we describe a compact data structure for encoding such a model. We also propose a new data structure for two-dimensional simplicial meshes, capable of representing both connectivity and adjacency information with a small memory overhead, which is used to describe the mesh extracted from an NMT through selective refinement. Finally, we present algorithms to efficiently perform updates on such a data structure.