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Undergraduate
By
Shiva Kintali
on June 25, 2012 | Updated Dec. 6, 2017
Separator number of a tree
Let \(T(V,E)\) be a tree with \(|V| = n\) vertices. Let \(W\) be a subset of \(V\). Prove that there is a vertex \(v \in V\) such that every component of \(T - v\) contains at most …
Computer Science
Mathematics
Algorithms
Graph Theory
trees
0
Undergraduate
By
Shiva Kintali
on June 17, 2012 | Updated Dec. 6, 2017
Trees in a graph
Let \(k \geq 1\) be an integer, and let \(T\) be a tree on \(k+1\) vertices. Show that if a graph \(G\) has minimum degree at least \(k\), then \(G\) has a subgraph isomorphic to \(T\).
Mathematics
Graph Theory
trees
0
Undergraduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Subtrees of a Tree
Let \(T_1, T_2, \dots, T_k\) be subtrees of a tree \(T\). Prove that if every two of them have a vertex in common, then they all have a vertex in common.
Mathematics
Graph Theory
trees
0
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Minimum spanning trees
Suppose we are given a connected graph \(G\) with weights on edges, such that all edge weights are distinct. For any cycle in \(G\), prove that the maximum weight edge in the cycle cannot be in a…
Computer Science
Mathematics
Algorithms
Graph Theory
graph cut
spanning tree
trees
0
Undergraduate
By
Shiva Kintali
on Oct. 3, 2013 | Updated Jan. 4, 2018
Leaves in a tree
Prove that every tree has at least two leaves (i.e., vertices of degree \(1\)). Prove that every tree with maximum degree \(\Delta>1\) has at least \(\Delta\) leaves.
Mathematics
Graph Theory
counting
trees
0
Undergraduate
By
Shiva Kintali
on Oct. 4, 2013 | Updated Jan. 4, 2018
Trees and non-planarity
Let \(V\) be a set of vertices, and let \(T_1=(V,E_1)\), \(T_2=(V,E_2)\), and \(T_3=(V,E_3)\) be three trees on the vertices of \(V\) with disjoint sets of edges: …
Mathematics
Graph Theory
planar graphs
trees
0
Undergraduate
By
Shiva Kintali
on June 6, 2013 | Updated Dec. 6, 2017
There are exactly _________ spanning trees on the vertex set {\(1, \dots , n\)}.
Mathematics
Graph Theory
counting
trees
0
Undergraduate
By
Shiva Kintali
on July 2, 2012 | Updated Dec. 6, 2017
Linear time tree isomorphism
Let \(T_1 (V_1,E_2)\) and \(T_2(V_2,E_2)\) be two undirected unlabeled trees. Design a linear-time algorithm to decide if \(T_1\) is isomorphic to \(T_2\).
Computer Science
Mathematics
Algorithms
Graph Theory
graph isomorphism
linear time algorithms
trees
0
Undergraduate
By
Shiva Kintali
on July 10, 2012 | Updated Dec. 6, 2017
Diameter of a tree
Let \(T(V, E)\) be a tree given as an adjacency list. For vertices \(u, v \in V\), let \(d(u, v)\) denote the length of the path from \(u\) to \(v\) in \(T\). Give a linear-time algorithm to determine…
Computer Science
Mathematics
Algorithms
Graph Theory
linear time algorithms
trees
0
Undergraduate
By
Shiva Kintali
on June 8, 2012 | Updated Dec. 6, 2017
Linear time algorithms on trees
Let \(T(V,E)\) be a tree. Design linear time (i.e., \(O(|V|)\) time) algorithms for the following problems : Find an optimal vertex cover in \(T\). Find a maximum matching in \(T\). Find a maximum i…
Computer Science
Mathematics
Algorithms
Graph Theory
independent set
linear time algorithms
matching
trees
vertex cover
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