Graph Theory

# Graph Theory by M. Borowiecki, J.W. Kennedy, M.M. Syslo

By M. Borowiecki, J.W. Kennedy, M.M. Syslo

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Extra resources for Graph Theory

Example text

Chap. 3 Connectivity 49 Proof. Because H is a subgraph of G[W], every path in H is a path in G[W]. Moreover, for any w e W, every path in H — w is a path in G[W] — w. This proves part (i). If H is a block of G, then, by maximality and part (i), H = G[W], proving part (ii). If G[W] is a block, then part (iii) follows from part (i). Otherwise, it must be that G[W] fails to be maximal, so there is a connected subgraph H\ = (W\, Fi) of G such that H\ has no cut-vertices, G[W] is a subgraph of H\, and o(Wi) > o(W).

Let B = B(G) be the k x (n — k) matrix whose (i, y')-entry is 1 if {v¡, Vk+j) € £, and 0 otherwise. Then the adjacency matrix of G can be expressed (in "partitioned" form) as *G)=(£ J). (15) where each "0" represents a square submatrix consisting entirely of zeros and B' is the transpose of B; that is, B' is the (n — k) x k matrix whose (1, y)-entry is the (j, i)-entry of B. Conversely, if some numbering of the vertices of G produces an adjacency matrix of the form given in Equation (15), then G is bipartite.

A" 2 2 /4 if n is even, 2 if n is odd. -D/4 18 Chap. 1 Invariants 21 Let G be a connected graph on n > 2 vertices. Suppose G does not have an induced subgraph isomorphic to At or C4. Prove that G has a dominating vertex, in other words, that A(G) = n — 1. 22 A multigraph consists of two things: (1) a nonempty (vertex) set V and (2) a multiset E with the property that every element of £ is an element of V(2). So, a multigraph is like a graph except that more than one edge can be incident to the same pair of vertices.