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Each 1 represents a pair that is an edge, each 0 represents one that isn’t an edge. Thus Θ(n2 ) bits describe a graph. Since n2 is a polynomial in n, any function of the number of input data bits that can be bounded by a polynomial in same, can also be bounded by a polynomial in n itself. Hence, in the case of graph algorithms, the ‘easiness’ vs. ‘hardness’ judgment is not altered if we base the distinction on polynomials in n itself, rather than on polynomials in the number of bits of input data.

How many maximal independent sets does G have? 14. Describe the complement of the graph G in exercise 13 above. How many cliques does it have? 15. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, 3} a triangle? 16. Let H be a labeled graph of L vertices. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, . . , L} equal to H? 17. Devise an algorithm that will decide if a given graph, of n vertices and m edges, does or does not contain a triangle, in time O(max(n2 , mn)).

Then there is no way to color the vertices without ever finding that both endpoints of some edge have the same color. On the other hand, if we have four colors available then we can do the job. Fig. 4 There are many interesting computational and theoretical problems in the area of coloring of graphs. Just for its general interest, we are going to mention the four-color theorem, and then we will turn to a study of some of the computational aspects of graph coloring. First, just for general cultural reasons, let’s slow down for a while and discuss the relationship between graph colorings in general and the four-color problem, even though it isn’t directly relevant to what we’re doing.