Preliminary Notation Sample Clauses

Preliminary Notation. In this section, we introduce the notation we use and we assume that the reader is fa- miliar with the standard graph terminology, as contained for instance in [8]. We consider finite undirected bipartite graphs without loops or multiple edges. Bipartite graphs arise naturally in many other applications besides the ones addressed in this thesis since they enable to model the relations between two different classes of objects. For each of the graph problems in this thesis, we denote by n the number of nodes and by m the number of edges of the input graph. Given a bipartite graph G = (U ∪ W, E) and a node u ∈ U , we denote by NG(u) the set of nodes adjacent to u in G, u excluded, and by EG(u) the set of edges incident to u in G. When it is clear the graph G in which the neighbourhood is taken, we may omit it from the notation simply writing N (u) or E(u) respectively. Moreover, given Uj ⊆ U and Wj ⊆ W , we denote by G[Uj,Wj] the subgraph of G induced by Uj ∪ Wj. A node u ∈ U such that NG(u)= W is called a universal node. Chain Graphs A central role in this part of the thesis is played by chain graphs (also called difference graphs). As we prove in Section 1.4, they are related to the graph interpretation of the quan- titative model Toxin and Antitoxin (presented in Section 1.2). Although chain graphs are interesting in themselves as graph theoretical objects, they appear in many other applications. Indeed, not only are they related to other important graph classes such as threshold graphs [49] and interval graphs [91], but they can also be directly interpreted as the comparability graph of height one partial orders. Hereafter, we introduce chain graphs and some of their properties that will be used in this thesis.
Sample 1
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