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Unformatted text preview: Rob Bayer Math 55 Worksheet August 11, 2009 Spanning Trees 1. Find a spanning tree in each of the following graphs (C) 2. Find a spanning tree for
(a) K4 00) Q3 (C) 05
3. How many nonisomorphic spanning trees do each of the following graphs have?
(a) K3 (b) 05 (<3) is 4. Prove that if e is a cut edge of G, then 6 must be in every spanning tree for G 5. A forrest is an acyclic graph (ie, multiple trees). A spanning forrest for a simple graph G is a subgraph of G
that is a forrest, contains all the vertices of G, and has the same number of connected components as G. (a) Show that every simple graph has a spanning fon'est. (b) How many edges are there in the spanning forrest for a graph with n vertices and 6 connected
componenets? 6. Suppose T1 and T2 are two different spanning trees of the simple connected graph G. Show that if 31 is an edge
in T1 that is not in T2, then there is an edge 62 6 T2 such that T1 and T2 are both still spanning trees after you swap which T; each eg belongs to (ie, replace 81 with 82 in T1 and vice—versa). 7. A rooted spanning tree for a directed graph is a rooted tree containing edges of the graph such that every
vertex of G is the endpoint of an edge in T and all edges point “away” from the root. I (a) Give an example of a directed graph with no rooted spanning tree (b) Show that a connected digraph in which all vertices have the same in—degree as outdegree must contain a
rooted spanning tree ' Minimum Spanning Trees 1. Find a Minimum Spanning Tree (MST) for each of the following weighted graphs Q Q 2. What does an MST represent if the nodes of your graphs represent computers, the edges represent network
links, and the weights represent the cost to build those links? 3. A minimum spanning forrest for a simple weighted graph G 15 a spanning forrest (see above) of minimal total
weight. Show that every simple weighted graph has a minimal spanning forrest.  A or“, 4. Go back and do problem 6 from above if you haven’t yet. (a) Prove that if all the weights in G are different, then there is only one MST for G (b) Prove that every MST for a simple graph must contain both the smallest and second—smallest weight
edges. (c) Give an example of a graph G where the 3rd smallest edge is not a part of any MST. (d) Prove that if all edges incident to the vertex i) have different weights, then every MST must contain the
one of minimal weight. 5. The Traveling Salesman problem is a very famous problem in Computer Science and is as follows: Given a
complete, weighted graph on n vertices (ie, it looks like Kn with weights), ﬁnd a Hamiltonian Cycle (a cycle
that visits each node exactly once) of minimum total weight. Note: this problem gets its name from the
situation where you think of the graph as being a map of cities and the travel times between them. (a) How many Hamiltonian Cycles are there in such graph?
(b) Show that there must be one of minimal (but not necessarily least!) weight (c) Explain why both Prim’s and Kruskal’s Algorithms are not helpful in ﬁnding it
6. Prove that every connected graph on n vertices has at least it — 1 edges
' 7. So far, we’ve only been studying ﬁnite graphs. Let’s see which results still hold in inﬁnite graphs. (a) Show that if G has countably many vertices, then it also has countably many edges (b) Show that a connected inﬁnite graph has a spanning tree (c) What falls apart in our proof that every connected weighted ﬁnite graph has a minimal spanning tree?
(d) Show that if all edge weights are integers, then a minimal spanning tree is still guaranteed to exist. (e) Consider the graph G that has a vertex for each real number and an edge between the vertices
representing :1: and y if]? x — yl E N. Is this graph connected? How many connected components does it have? How many vertices are in each connected component? ...
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