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It should bc noted, however, that there exists a basis for loops as well as that for cutsets which is not a primitive set associated with a tree-cotree pair. For example, in Fig. 3(a), 51 = b, - b2, 52 = b2 - b3, 53 = b3 - b, is a nonprimitive basis for loops, and, in Fig. 3(b), 91 = b, - b,, 9 2 = b2 - b3, q 3 = b3 - b4 is a nonprimitive basis for cutsets. ) It is also worthwhile to note that a maximal independent set is not necessarily a basis. For example, in the graph of Fig.

A= 1 Choose M - rn nodes nu,, . . , nai E W i(i = 1, . , M - r n ) . ,E~ b,J. 10) We observe that the value of la is determined independently of the choice of the path from no to one of nai’s. For, if there is another path ( E l ’ b,,, . ,Eh’bA,,) and rl= h E i ? 11) 48 111. , el bKt, -eh'bAh, branches of a closed path, and . . 12) is a loop. 13) n so that 5,' = l a . Let us consider 'the cutset Tj = C 4, b, = Sc with 5 K=l thus determined. 14) If (cl bKl, . . , el bKJ is (the sequence of branches of) a path from nB to an n a i , then (b,, el bKl, .

It should bc noted, however, that there exists a basis for loops as well as that for cutsets which is not a primitive set associated with a tree-cotree pair. For example, in Fig. 3(a), 51 = b, - b2, 52 = b2 - b3, 53 = b3 - b, is a nonprimitive basis for loops, and, in Fig. 3(b), 91 = b, - b,, 9 2 = b2 - b3, q 3 = b3 - b4 is a nonprimitive basis for cutsets. ) It is also worthwhile to note that a maximal independent set is not necessarily a basis. For example, in the graph of Fig.

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