Bipartite Graphs and Subgraphs
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Bipartite Graphs and Subgraphs

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Questions and Answers

What is an edge cut of a graph G?

  • A set of edges where the resulting subgraph is connected
  • A set of edges that connects all vertices in G
  • A set of edges that includes all edges of G
  • A set of edges that can be removed to disconnect G (correct)

    • What does the edge connectivity of a graph G, denoted by λ(G), represent?

  • The maximum number of edges that can be removed without disconnecting the graph
  • The minimum number of edges in an edge cut of G (correct)
  • The total number of edges in the graph
  • The minimum number of vertices that can disconnect the graph

    • Which statement is true regarding strong connectedness in directed graphs?

  • Strongly connected graphs can be disconnected by removing one edge.
  • There are no paths between any two vertices.
  • There must be a single path from any vertex to every other vertex.
  • There is a path from vertex a to b and from b to a for all vertices a and b. (correct)

    • What defines a weakly connected directed graph?

    <p>There is a path between every two vertices in the underlying undirected graph.</p> Signup and view all the answers

    Which of the following relationships is correct regarding the edge connectivity of a graph G?

    <p>κ(G) ≤ λ(G) ≤ δ(G)</p> Signup and view all the answers

    What is the defining characteristic of a bipartite graph?

    <p>Vertices can be divided into two disjoint sets with edges only between the sets.</p> Signup and view all the answers

    Which of the following can be a complete bipartite graph?

    <p>All of the above</p> Signup and view all the answers

    Which of the following graphs is NOT bipartite?

    <p>C5 (cycle with 5 vertices)</p> Signup and view all the answers

    What is the relationship between subgraphs and proper subgraphs?

    <p>A proper subgraph cannot equal the parent graph.</p> Signup and view all the answers

    How can a simple graph's vertices be colored in order to verify if it is bipartite?

    <p>Two different colors must be used without adjacent vertices sharing the same color.</p> Signup and view all the answers

    Which of the following statements about bipartite graphs is true?

    <p>Every cycle in a bipartite graph is an even cycle.</p> Signup and view all the answers

    What does it mean for a bipartite graph to be complete?

    <p>Every vertex in one set is connected to every vertex in the other set.</p> Signup and view all the answers

    In the context of graph theory, what does a subgraph induced by a vertex subset represent?

    <p>A graph that includes only the selected vertices and connections between them.</p> Signup and view all the answers

    What characterizes a regular graph?

    <p>All vertices have the same degree.</p> Signup and view all the answers

    For which type of graph is every vertex guaranteed to have the same degree?

    <p>Regular graph</p> Signup and view all the answers

    Which of the following statements about trails is correct?

    <p>A trail is defined by having no repeated edges.</p> Signup and view all the answers

    What is the degree sequence of a complete graph K5?

    <p>(4, 4, 4, 4, 4)</p> Signup and view all the answers

    Which type of graph is guaranteed to be bipartite?

    <p>Tree graph</p> Signup and view all the answers

    What is a characteristic of a closed trail?

    <p>It starts and ends at the same vertex.</p> Signup and view all the answers

    What is the result of removing a vertex v and all edges incident to it from graph G?

    <p>A subgraph denoted by G - v</p> Signup and view all the answers

    What characterizes the union of two simple graphs G1 and G2?

    <p>It results in a graph with vertex set V1 ∪ V2 and edge set E1 ∪ E2</p> Signup and view all the answers

    What is a requirement for two simple graphs to be considered isomorphic?

    <p>They must have the same number of vertices, edges, and vertex degrees</p> Signup and view all the answers

    Which term describes a simple graph that is isomorphic to its own complementary graph?

    <p>Self-complementary graph</p> Signup and view all the answers

    What does the degree sequence of a graph represent?

    <p>The count of edges incident to each vertex, usually in nonincreasing order</p> Signup and view all the answers

    According to the Havel Hakimi Theorem, under what condition is an integer list d graphic?

    <p>If the largest element can be deleted and conditions on remaining elements are satisfied</p> Signup and view all the answers

    What is implied if a sequence is termed as graphic?

    <p>It is the degree sequence of some simple graph</p> Signup and view all the answers

    In graph theory, what can be concluded if two simple graphs are not isomorphic?

    <p>They must have different numbers of vertices</p> Signup and view all the answers

    What is a connected component of a graph?

    <p>A connected subgraph not included in another connected subgraph</p> Signup and view all the answers

    Which of the following statements about cut vertices is true?

    <p>Removing a cut vertex creates a disconnected subgraph.</p> Signup and view all the answers

    What characterizes a nonseparable graph?

    <p>It cannot be disconnected by removing any vertex.</p> Signup and view all the answers

    In the context of graph connectivity, what does the symbol κ(G) represent?

    <p>The minimum number of vertices needed to disconnect graph G</p> Signup and view all the answers

    Which of the following graphs cannot have cut vertices?

    <p>A complete graph K3</p> Signup and view all the answers

    What is true about a 2-connected graph?

    <p>It is connected and has no cut vertices.</p> Signup and view all the answers

    What is the definition of a cut edge in graph theory?

    <p>An edge whose removal increases the number of connected components</p> Signup and view all the answers

    How is a graph described if it is k-connected?

    <p>If it requires at least k vertices to be removed to disconnect it</p> Signup and view all the answers

    Study Notes

    Bipartite Graphs

    • A Bipartite graph is a graph that can be divided into two distinct sets (V1 and V2) of vertices such that every edge connects a vertex in V1 to a vertex in V2, never connecting two vertices in the same set.
    • A simple graph is bipartite if and only if its vertices can be colored with two distinct colors without any adjacent vertices having the same color.

    Complete Bipartite Graphs

    • A complete bipartite graph Km,n has two sets of vertices, one with 'm' vertices and the other with 'n' vertices.
    • Every vertex in the first set is connected to every vertex in the second set, resulting in a complete bipartite graph.

    Subgraphs

    • A subgraph of a graph G is a graph H whose vertices and edges are a subset of G.
    • An induced subgraph is created by selecting a set of vertices from the original graph and including all edges that connect those vertices in the original graph.
    • Removing a vertex 'v' and all its incident edges from a graph G produces a subgraph denoted by G - v.

    Union of two Graphs

    • The union of two graphs, G1 and G2, is a new graph with the combined vertex set of both graphs and their combined edge sets.
    • The complement of a graph G has the same vertices as G, but two vertices are adjacent in the complement only if they are not adjacent in the original graph.

    Graph Isomorphism

    • Two graphs are considered isomorphic if they have the same structure, even if their vertex labels or edge arrangement are different.
    • An isomorphism between two graphs is a one-to-one correspondence between the vertices of the two graphs that preserves adjacency relationships.
    • Key features for two graphs to be isomorphic:
      • They must have the same number of vertices.
      • They must have the same number of edges.
      • The degrees of corresponding vertices in both graphs must be the same.

    Graphic Sequence

    • The degree sequence of a graph is the list of degrees of all vertices, usually written in non-increasing order.
    • A graphic sequence is a sequence that can be the degree sequence of a simple graph.
    • Not all sequences are graphic, and the Havel-Hakimi Theorem can be used to determine if a sequence is graphic.
    • In a simple graph with at least two vertices, there must be at least two vertices with the same degree.
    • Regular graphs are graphs where all vertices have the same degree, and an n-regular graph has a degree of n for every vertex.
    • Examples of regular graphs are Kn, Cn, Wn, and Qn for specific values of 'n'.
    • A walk in a graph is an alternating sequence of vertices and edges, starting and ending with vertices.
    • A closed walk is a walk that starts and ends at the same vertex.
    • A trail is a walk with no repeated edges.
    • A closed trail is called a circuit.
    • A path is a trail with no repeated vertices.

    Connected Graph

    • An undirected graph is called connected if there is a path between every pair of distinct vertices.
    • A disconnected graph is a graph that is not connected.
    • Every pair of distinct vertices in a connected graph has a simple path between them.

    Components of a Graph

    • A connected component of a graph is a subgraph that is connected and not a proper subgraph of another connected subgraph.
    • A connected component of a graph is a maximal connected subgraph, meaning it cannot be expanded further.

    Cut vertex and Cut edge

    • A cut vertex (or articulation point) is a vertex whose removal increases the number of connected components in a graph.
    • A cut edge (or bridge) is an edge whose removal increases the number of connected components in a graph.
    • If a graph has more than two vertices, at least one endpoint of a bridge is a cut vertex.

    Nonseparable Graphs

    • A connected graph without a cut vertex is called a nonseparable graph.
    • The complete graph Kn, where n ≥ 3, has no cut vertices.
    • Connectivity in graphs refers to the number of vertices or edges needed to be removed to disconnect the graph.
    • A vertex cut (or separating set) is a subset of vertices whose removal disconnects the graph.
    • The vertex connectivity of a graph is the minimum number of vertices in a vertex cut.

    k-connected graph

    • A graph is k-connected if it cannot be disconnected by removing fewer than 'k' vertices.
    • A graph is 1-connected if it is connected and not a single vertex graph.
    • A graph is 2-connected or biconnected if it is nonseparable and has at least three vertices.

    Edge Connectivity

    • The edge connectivity of a graph is the minimum number of edges that need to be removed to disconnect the graph.
    • The edge connectivity of a graph is always less than or equal to the minimum degree of the graph.

    Connectedness in Directed Graphs

    • A directed graph is strongly connected if there is a path from any vertex to any other vertex in the graph, going in either direction.
    • A directed graph is weakly connected if the underlying undirected graph (ignoring directions) is connected.

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    02 Graph Theory - II PDF

    Description

    This quiz explores the concepts of bipartite graphs, complete bipartite graphs, and subgraphs. You'll learn about the distinct sets of vertices in bipartite graphs and how to identify and create subgraphs. Test your understanding of these fundamental graph theory topics.

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