Chapter 11: Problem 1
List three situations, different from those in this section, where a graph could prove useful.
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These are the key concepts you need to understand to accurately answer the question.
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Chapter 11: Problem 1
List three situations, different from those in this section, where a graph could prove useful.
These are the key concepts you need to understand to accurately answer the question.
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If \(G=(V, E)\) is an undirected graph, a subset \(D\) of \(V\) is called a dominating set if for all \(v \in V\), either \(v \in D\) or \(v\) is adjacent to a vertex in \(D\). If \(D\) is a dominating set and no proper subset of \(D\) has this property, then \(D\) is called minimal. The size of any smallest dominating set in \(G\) is denoted by \(\gamma(G)\) and is called the domination number of \(G\). a) If \(G\) has no isolated vertices, prove that if \(D\) is a minimal dominating set, then \(V-D\) is a dominating set. b) If \(I \subseteq V\) is independent, prove that \(I\) is a dominating set if and only if \(I\) is maximal independent. c) Show that \(\gamma(G) \leq \beta(G)\), and that \(|V| \leq \beta(G) \chi(G)\). [Here \(\beta(G)\) is the independence number of \(G-\) first given in Exercise 25 of Section 11.5.]
Let \(G=(V, E)\) be a loop-free connected graph with \(|V|=\) \(v\). If \(|E|>(v / 2)^{2}\), prove that \(G\) cannot be bipartite.
Let \(G=(V, E)\) be a loop-free undirected graph. Prove that if \(G\) contains no cycle of odd length, then \(G\) is bipartite.
a) Let \(G=(V, E)\) be a loop-free undirected graph. Recall that \(G\) is called self-complementary if \(G\) and \(\bar{G}\) are isomorphic. If \(G\) is self- complementary (i) determine \(|E|\) if \(|V|=n\); (ii) prove that \(G\) is connected. b) Let \(n \in \mathbf{Z}^{+}\), where \(n=4 k\left(k \in \mathbf{Z}^{+}\right)\)or \(n=4 k+1\) \((k \in \mathbf{N})\), Prove that there exists a self-complementary graph \(G=(V, E)\), where \(|V|=n\)
a) If \(G=(V, E)\) is an undirected graph with \(|V|=v\), \(|E|=e\), and no loops, prove that \(2 e \leq v^{2}-v\) b) State the corresponding inequality for the case when \(G\) is directed.
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