Lecture 4: Walks, Trails, Paths and Connectivity

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1 Lecture 4: Walks, Trails, Paths and Connectivity Rosa Orellana Math 38 April 6, 2015

2 Graph Decompositions Def: A decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list.

3 Graph Decompositions Def: A decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. Definition: An n-vertex graph H is self-complementary if it is possible to decompose K n into two copies of H.

4 Walks, Trails, Paths and Cycles Def: A walk is an alternating list v 0, e 1, v 1, e 2, v 2,..., e k, v k of vertices and edges such that for 1 i k the edge e i has endpoints v i 1 and v i. An u, v-walk is a walk with first vertex u and last vertex v.

5 Walks, Trails, Paths and Cycles Def: A walk is an alternating list v 0, e 1, v 1, e 2, v 2,..., e k, v k of vertices and edges such that for 1 i k the edge e i has endpoints v i 1 and v i. An u, v-walk is a walk with first vertex u and last vertex v. Def: A trail is a walk with no repeated edge. An u, v-trail is a trail with first vertex u and last vertex v.

6 Walks, Trails, Paths and Cycles Def: A walk is an alternating list v 0, e 1, v 1, e 2, v 2,..., e k, v k of vertices and edges such that for 1 i k the edge e i has endpoints v i 1 and v i. An u, v-walk is a walk with first vertex u and last vertex v. Def: A trail is a walk with no repeated edge. An u, v-trail is a trail with first vertex u and last vertex v. Def: A path is a trail with no repeated vertex. An u, v-path is a path with first vertex u and last vertex v.

7 Walks, Trails, Paths and Cycles Def: A walk is an alternating list v 0, e 1, v 1, e 2, v 2,..., e k, v k of vertices and edges such that for 1 i k the edge e i has endpoints v i 1 and v i. An u, v-walk is a walk with first vertex u and last vertex v. Def: A trail is a walk with no repeated edge. An u, v-trail is a trail with first vertex u and last vertex v. Def: A path is a trail with no repeated vertex. An u, v-path is a path with first vertex u and last vertex v. Def: If a walk or trail have the same endpoints u = v then we say they are closed.

8 Walks, Trails, Paths and Cycles Def: A walk is an alternating list v 0, e 1, v 1, e 2, v 2,..., e k, v k of vertices and edges such that for 1 i k the edge e i has endpoints v i 1 and v i. An u, v-walk is a walk with first vertex u and last vertex v. Def: A trail is a walk with no repeated edge. An u, v-trail is a trail with first vertex u and last vertex v. Def: A path is a trail with no repeated vertex. An u, v-path is a path with first vertex u and last vertex v. Def: If a walk or trail have the same endpoints u = v then we say they are closed. Def: A closed trail is called a circuit and a circuit with no repeated vertex is called a cycle.

9 Some comments on the definitions Remarks: A path cannot be closed by definition.

10 Some comments on the definitions Remarks: A path cannot be closed by definition. For simple graphs we do not have to write the edges in the above definitions, only the vertices suffice because there is only one edge between any two vertices.

11 Some comments on the definitions Remarks: A path cannot be closed by definition. For simple graphs we do not have to write the edges in the above definitions, only the vertices suffice because there is only one edge between any two vertices. When we write cycles we will use parenthesis to denote it without repeating the first vertex.

12 Some comments on the definitions Remarks: A path cannot be closed by definition. For simple graphs we do not have to write the edges in the above definitions, only the vertices suffice because there is only one edge between any two vertices. When we write cycles we will use parenthesis to denote it without repeating the first vertex. Def: The length of a cycle, path, trail, or walk is the number of edges transversed.

13 Some comments on the definitions Remarks: A path cannot be closed by definition. For simple graphs we do not have to write the edges in the above definitions, only the vertices suffice because there is only one edge between any two vertices. When we write cycles we will use parenthesis to denote it without repeating the first vertex. Def: The length of a cycle, path, trail, or walk is the number of edges transversed. Lemma: Every u, v-walk contains an u, v-path.

14 Components Def: The components of a graph G are its maximal connected subgraphs.

15 Components Def: The components of a graph G are its maximal connected subgraphs. Proposition: Every graph with n vertices and k edges has at least n k components.

16 Components Def: The components of a graph G are its maximal connected subgraphs. Proposition: Every graph with n vertices and k edges has at least n k components. Note: A vertex of degree 0 is called an isolated vertex. It is a connected component on its own, called trivial connected component.

17 Cut-edge, cut-vertex Def: A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components.

18 Cut-edge, cut-vertex Def: A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. Notation: G e is the graph obtained by deleting the edge e from G. Note that when we delete an edge we do not remove its endpoints.

19 Cut-edge, cut-vertex Def: A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. Notation: G e is the graph obtained by deleting the edge e from G. Note that when we delete an edge we do not remove its endpoints. G v is the graph obtained by deleting the vertex v from G. Note that when we delete a vertex we delete all the edges adjacent to it.

20 Cut-edge, cut-vertex Def: A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. Notation: G e is the graph obtained by deleting the edge e from G. Note that when we delete an edge we do not remove its endpoints. G v is the graph obtained by deleting the vertex v from G. Note that when we delete a vertex we delete all the edges adjacent to it. G S is the graph obtained by deleting the vertices in the set S V (G).

21 Cut-edge, cut-vertex Def: A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. Notation: G e is the graph obtained by deleting the edge e from G. Note that when we delete an edge we do not remove its endpoints. G v is the graph obtained by deleting the vertex v from G. Note that when we delete a vertex we delete all the edges adjacent to it. G S is the graph obtained by deleting the vertices in the set S V (G). G M is the graph obtained by deleting the edges in the set M E(G).

22 Induced Subgraphs Def: An induced subgraph is the subgraph obtained by deleting a set of vertices.

23 Induced Subgraphs Def: An induced subgraph is the subgraph obtained by deleting a set of vertices. Remarks: If T V (G) then T = V (G) T is its complement.

24 Induced Subgraphs Def: An induced subgraph is the subgraph obtained by deleting a set of vertices. Remarks: If T V (G) then T = V (G) T is its complement. G[T ] = G T is the induced graph obtained by removing the vertices in T.

25 Induced Subgraphs Def: An induced subgraph is the subgraph obtained by deleting a set of vertices. Remarks: If T V (G) then T = V (G) T is its complement. G[T ] = G T is the induced graph obtained by removing the vertices in T. G[T ] is the graph with vertex set T and all edges in G that connect vertices in T.

26 Characterization of cut-edges Theorem: cycle. An edge is a cut-edge if and only if it belongs to no

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