Here are the Comments on Problem Set 1.

The average score for PS1 is 60 (full-credit answers for all non-bonus questions would be worth 78 total points). By question, the averages were:

1a 1.96/2 1b 1.98/2 1c 1.95/2 1d 1.85/2 1e 0.54/2 2 9.91/10 3 3.98/10 5 9.49/10 6 8.31/10 7 5.16/10 8 9.07/10 bonus 1.71 9 5.39/8

According to Sipser, a graph is a tree if it is connected and has no simple cycles. Furthermore, a simple cycle is one that contains at least three nodes and repeats only the first and last nodes.

If we go by Sipser’s definitions of tree and simple cycle, are connected graphs containing cycles that are

notsimple cycles still considered trees?A tree cannot have any cycles. Excluding self-edges, if a graph has a cycle it must have a simple cycle. We can always remove extra repeated sequence from the cycle until it becomes a simple cycle.

I think you are right, though, that Sipser’s definition may not be quite as precise as it should be, since a graph with a self-edge has a cycle but may not have a simple cycle. Sipser’s definition would consider such a graph a tree, but I don’t think many people would agree with this (and I don’t think it was his intent, since none of the example graphs before this exhibit self-edges). A better (and simpler) definition of a tree is a fully-connected graph with no cycles. Or, a graph where every pair of vertices are connected by exactly on simple path. I’m not sure why Sipser’s definition unnecessarily over-specifies that the graph has no simple cycles, instead of just that it has no cycles.

I think Sipser’s definition of a graph, “a set of points with lines connecting some of the points”, is not quite precise enough but could be interpreted to not allow self-edges. If this is the case, the graph described in PS1, Problem 3 is not actually a graph according to this definition! The standard definition of a “simple graph” does disallow self edges, since each edge is a two-element set of vertices. If you answered question 3 correctly based on this definition of tree (that is, excluding the self-edges from G to make it a simple graph, or equivalently counting graphs with no simple cycles as trees even if they have self-cycles since this would be completely justified by Sipser’s definition) then you should get full credit for this question.