Show that a graph with at least 2k vertices is k-connected if and
only if for...
Show that a graph with at least 2k vertices is k-connected if and
only if for any two unrelated subsets X, Y of V, such that | X | =
k = | Y |, there are k foreign paths between X and Y.
Solutions
Expert Solution
since, a graph is k connected if and only if it has k disjoint
paths joining two arbitrary vedrtex sets say X and Y of size k each
of many to many typeof which a vertex belongs to both the sets is
valid as one vertex path.
GRAPH THEORY
Prove/Show that a connected Graph G is not separable if
and only if it is nonseparable.
Definitions for Reference: A connected Graph G is called
nonseparable if it has no cut vertices (A vertex v in a connected
graph G is caled a cut vertex if G-v is disconnected)
A connected graph G is called separable if there exist subgraphs
H1, H2 ⊂ G. with E(H1) ∪ E(H2) = E(G) and E(H1) ∩ E(H2) = ∅. V (H1)...
Given a connected graph G with n vertices. We say an edge of G
is a bridge if the graph becomes a disconnected graph after
removing the edge. Give an O(m + n) time algorithm that finds all
the bridges. (Partial credits will be given for a polynomial time
algorithm.) (Hint: Use DFS)
Let k be an integer satisfying k ≥ 2. Let G be a connected graph
with no cycles and k vertices. Prove that G has at least 2 vertices
of degree equal to 1.