MCA 520: Graph Theory Instructor Neelima Gupta
Description: MCA 520: Graph Theory Instructor Neelima Gupta nguptacs.du.ac.in Table of Contents Walks, Trails and Paths Walks May have repeated Edges and Vertices. In case of multi-graph, we include the edges also. In a simple graph, we can omit the
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slide1. MCA 520: Graph Theory Instructor
Neelima Gupta
ngupta@cs.du.ac.in<br>
slide2. Table of Contents Walks, Trails and Paths<br>
slide3. Walks May have repeated Edges and Vertices.
In case of multi-graph, we include the edges also. In a simple graph, we can omit the edges and simply mention the sequence of vertices.
Closed Walk<br>
slide4. Trails No repeated edges but vertices may repeat.
Closed Trail<br>
slide5. Path Neither vertices nor edges repeat.
Definition:
we say that a u-v walk W contains a u-v path P if all the edges and vertices of P occur in W and in that order but not necessarily consecutive.
Similarly a closed walk W contains a cycle C if …..
Lemma: Every u-v walk contains a u-v path<br>
slide6. Odd/Even Walk Odd/Even walk : number of edges is odd/even
Lemma: Every closed odd walk contains an odd cycle.
Remark: A closed even walk need not even contain a cycle, it may simply repeat edges. But, if an edge e appears exactly once in a closed walk, then the walk contains a cycle through e.<br>
slide7. Even Graph A vertex is stb even(/odd) if its degree is even(/odd).
A graph is stb an even graph if all its vertices are even.<br>
slide8. Maximal Path A path in a graph is stb maximal if it is not contained in a longer path.
If a graph is finite, maximal path always exists.
If every vertex in a finite graph G has degree at least 2 then it contains a cycle.
This is not true if the graph is not finite.<br>
slide9. Connection Relation (u,v): u is stb connected to v ……
Symmetric, Reflexive, Transitive
Equivalence Relation
Equivalence Class: Connected Component<br>
slide10. Lemma: A graph with n vertices and k edges has at least n – k components.
Proof: A graph with no edges has n components. Adding an edges reduces the number of components by at most 1. Thus after adding k edges, number of components is at least n – k.<br>
slide11. Deleting an edge/vertex G – e: Deleting an edge does not delete its incident vertices.
G – v: Deleting a vertex delete its incident edges.
Thus deleting an edge may increase the number of components by at most 1.
Deleting a vertex v may increase the number of components by (more) at most deg(v) – 1.
Induced Graph G[T] = Graph that remains after deleting some vertices such that the set of remaining vertices is T. i.e.
G[T] = (T, E(T)), where E(T) = {(u, v):u,v are in T and (u,v) is an edge in G}
Every subgraph of a graph need not be an induced subgraph.<br>
slide12. Cut-edge and Cut-Vertex An edge e is stb a cut edge if …
A vertex v is stb a cut vertex if …
Characterize cut-edges in terms of cycles.
Theorem: An edge is a cut edge iff it does not belong to any cycle.<br>
slide13. Bi-partite Graphs Konig Theorem : Characterizing Bipartite Graphs in terms of cycles: A graph is bipartite iff it has no odd cycles.
Testing whether a graph is bipartite:<br>
slide14. Union of Graphs Definition:
K4 : a union of two 4-cycles.<br>
slide15. Kn can be expressed as a union of k bi-partite graphs iff n < 2k.<br>
slide16. Eulerian Circuits A graph is Eulerian if it has a closed trail containing all the edges.
A graph is Eulerian iff it has at most one non-trivial component and all its vertices have even degree.<br>
Neelima Gupta
ngupta@cs.du.ac.in<br>
slide2. Table of Contents Walks, Trails and Paths<br>
slide3. Walks May have repeated Edges and Vertices.
In case of multi-graph, we include the edges also. In a simple graph, we can omit the edges and simply mention the sequence of vertices.
Closed Walk<br>
slide4. Trails No repeated edges but vertices may repeat.
Closed Trail<br>
slide5. Path Neither vertices nor edges repeat.
Definition:
we say that a u-v walk W contains a u-v path P if all the edges and vertices of P occur in W and in that order but not necessarily consecutive.
Similarly a closed walk W contains a cycle C if …..
Lemma: Every u-v walk contains a u-v path<br>
slide6. Odd/Even Walk Odd/Even walk : number of edges is odd/even
Lemma: Every closed odd walk contains an odd cycle.
Remark: A closed even walk need not even contain a cycle, it may simply repeat edges. But, if an edge e appears exactly once in a closed walk, then the walk contains a cycle through e.<br>
slide7. Even Graph A vertex is stb even(/odd) if its degree is even(/odd).
A graph is stb an even graph if all its vertices are even.<br>
slide8. Maximal Path A path in a graph is stb maximal if it is not contained in a longer path.
If a graph is finite, maximal path always exists.
If every vertex in a finite graph G has degree at least 2 then it contains a cycle.
This is not true if the graph is not finite.<br>
slide9. Connection Relation (u,v): u is stb connected to v ……
Symmetric, Reflexive, Transitive
Equivalence Relation
Equivalence Class: Connected Component<br>
slide10. Lemma: A graph with n vertices and k edges has at least n – k components.
Proof: A graph with no edges has n components. Adding an edges reduces the number of components by at most 1. Thus after adding k edges, number of components is at least n – k.<br>
slide11. Deleting an edge/vertex G – e: Deleting an edge does not delete its incident vertices.
G – v: Deleting a vertex delete its incident edges.
Thus deleting an edge may increase the number of components by at most 1.
Deleting a vertex v may increase the number of components by (more) at most deg(v) – 1.
Induced Graph G[T] = Graph that remains after deleting some vertices such that the set of remaining vertices is T. i.e.
G[T] = (T, E(T)), where E(T) = {(u, v):u,v are in T and (u,v) is an edge in G}
Every subgraph of a graph need not be an induced subgraph.<br>
slide12. Cut-edge and Cut-Vertex An edge e is stb a cut edge if …
A vertex v is stb a cut vertex if …
Characterize cut-edges in terms of cycles.
Theorem: An edge is a cut edge iff it does not belong to any cycle.<br>
slide13. Bi-partite Graphs Konig Theorem : Characterizing Bipartite Graphs in terms of cycles: A graph is bipartite iff it has no odd cycles.
Testing whether a graph is bipartite:<br>
slide14. Union of Graphs Definition:
K4 : a union of two 4-cycles.<br>
slide15. Kn can be expressed as a union of k bi-partite graphs iff n < 2k.<br>
slide16. Eulerian Circuits A graph is Eulerian if it has a closed trail containing all the edges.
A graph is Eulerian iff it has at most one non-trivial component and all its vertices have even degree.<br>