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Although a walk is a sequence of edges, it naturally induces a sequence of adjacent vertices which the edges connect. So we may denote the walk (a0 a1, a1 a2, . . . , ak 1 ak) more simply by a0 a1 a2 . . . ak 1 ak as long as each pair (ai 1, ai) is a valid edge in the graph. If p = a0 a1 a2 . . . ak 1 ak is a walk in a graph, then we refer to p as a walk from a0 to ak (or from ak to a0), and we say that p connects a0 to ak and that a0 and ak are connected by p. We also refer to a0 and ak as the terminal points or the end points of the walk. A path is a walk whose vertices are all distinct. EXAMPLE 15.4 Graph Paths
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In the graph in Figure 15.4, abcfde is a path of length 5. It is, more formally, the path (ab, bc, cf, fd, de). The walk abefdbc of length 6 is not a path because vertex b appears twice. The walk abefa of length 4 is also not a path. The sequence abf is not a walk because bf is not an edge. And the sequence abb is not a walk because bb is not an edge. Finally, aba is a walk of length 2, and ab is a walk of length 1.
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A graph is said to be connected if every pair of its vertices are connected by some path. A graph that is not connected is called disconnected. All the graphs in the previous examples are connected. EXAMPLE 15.5 A Disconnected Graph
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Figure 15.5 shows a graph of size 12 that is not connected.
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Figure 15.4 Paths in graphs
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A connected component of a graph is a Figure 15.5 A disconnected graph subgraph that is maximally connected, that is, a connected subgraph with the property that any larger subgraph that contains it is disconnected. The graph in Example 15.5 has five connected components, of sizes 3, 1, 4, 2, and 2. Theorem 15.3 Every graph is a union of a unique set of connected components. A walk is closed if its two end points are the same vertex. A cycle is a closed walk of length at least 3 whose interior vertices are all distinct. EXAMPLE 15.6 Graph Cycles
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In the graph shown in Figure 15.4: The walk abefa is a cycle. The walk abedbcfa is not a cycle because it is not a path: It has the duplicate internal vertex b. The path abef is not a cycle because it is not closed. And the walk aba is not a cycle because its length is only 2.
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A graph is said to be acyclic if it contains no cycles. Among the graphs shown above, only the ones in Figure 15.2 on page 286 are acyclic. An acyclic graph is also called a free forest, and a connected acyclic graph is also called a free tree. Note that a tree is the same as a free tree in which one node has been designated as the root. So in the context of graph theory, a tree is called a rooted tree, which is defined to be a connected acyclic graph with one distinguished node. A spanning tree of a graph is a connected acyclic spanning subgraph. EXAMPLE 15.7 Spanning Trees
Figure 15.6 shows a graph and a spanning tree for it.
Figure 15.6 The graph on the right is a spanning tree of the graph on the left
ISOMORPHIC GRAPHS An isomorphism between two graphs G = (V, E) and G = (V , E ) is a function f that assigns to each vertex x in V some vertex y = f(x) in V so that the following three conditions are satisfied: 1. f is one-to-one: Each x in V gets assigned a different y = f(x) in V 2. f is onto: Every y in V gets assigned to some x in V. 3. f preserves adjacency: If {x 1, x 2} is an edge in E, then {f(x 1), f(x 2)} is an edge in E . Two graphs are said to be isomorphic if there is an isomorphism from one to the other. The word isomorphic means same form. When applied to graphs, it means that they have the same topological structure. Graphically, two graphs are isomorphic if one can be twisted around to the same shape as the other without breaking any of the edge connections. EXAMPLE 15.8 Isomorphic Graphs
The two graphs in Figure 15.7 on page 289 are isomorphic. The isomorphism is indicated by the corresponding vertex labels. It can be verified that if vertex x 1 is adjacent to vertex x 2 in one graph, then the corresponding vertices are adjacent in the other graph. For example, vertex a is adjacent to vertices b, d, e, and f (but not c, g, or h) in both graphs.
To prove that two graphs are isomorphic (by definition), it is necessary to find an isomorphism between them. This is equivalent to labeling both graphs with the same set of labels so that adjacency applies equally to both labelings. Finding such an isomorphism by chance is unlikely
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