Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 511 — #519
12.10. References 511
(c) Suppose G is edge-weighted, the weight of e is larger than the weights of all
the other edges, e is on a cycle in G, and e is an edge of T . Conclude that T is not
a minimum weight spanning tree of G.
Class Problems
Problem 12.57.
Procedure Mark starts with a connected, simple graph with all edges unmarked and
then marks some edges. At any point in the procedure a path that includes only
marked edges is called a fully marked path, and an edge that has no fully marked
path between its endpoints is called eligible.
Procedure Mark simply keeps marking eligible edges, and terminates when there
are none.
Prove that Mark terminates, and that when it does, the set of marked edges forms
a spanning tree of the original graph.
Problem 12.58.
A procedure for connecting up a (possibly disconnected) simple graph and creating
a spanning tree can be modelled as a state machine whose states are finite simple
graphs. A state is final when no further transitions are possible. The transitions are
determined by the following rules:
Procedure create-spanning-tree
1. If there is an edge hu—vi on a cycle, then delete hu—vi.
2. If vertices u and v are not connected, then add the edge hu—vi.
(a) Draw all the possible final states reachable starting with the graph with vertices
f1; 2; 3; 4g and edges
fh1—2i ; h3—4ig:
(b) Prove that if the machine reaches a final state, then the final state will be a tree
on the vertices of the agraph on which it started.
(c) For any graph G 0 , let e be the number of edges in G 0 , c be the number of connected
components it has, and s be the number of cycles. For each of the quantities
below, indicate the strongest of the properties that it is guaranteed to satisfy, no
matter what the starting graph is.