1/2-Transitive Graphs of Order 3p by Alspach B., Xu M.Y. PDF

By Alspach B., Xu M.Y.

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Extra info for 1/2-Transitive Graphs of Order 3p

Example text

Conversely, if we can verify this condition for every Ge, then it follows that R is in Robinson form. Speaking more generally, suppose G is an arbitrary graph and :< is an ordering of the vertices of G. We say that < is compatible with G if whenever / ^ / ^ k ^ / and there is an edge in G between / and /, then there is an edge in G between / and k. For example, in the graph of Fig. , if we assume a loop at each vertex, then a compatible ordering of the vertices is given by c, b, e, a, d. The ordering c, b, a, e, d is not compatible since b :< b ^ a ^ e and there is an edge between b and e, but no edge between b and a.

7 We shall suppress the loop at each vertex in this and all graphs of this section. Again, there is a (suppressed) loop at each vertex. INDIFFERENCE, MEASUREMENT, AND SERIATION 29 numbers f(a) and f(c) must be at least «5 apart. It follows that these numbers are FIG. 3. The argument that Z4 is not an indifference graph. arranged as in Fig. 3. Now f(d) must be within 8 of /(a) and /(c), but not within S of f(b). This is impossible. Thus, Z4 is not an indifference graph. Neither is Zn, any n ^4. 9 In particular, the graph of indifferences cannot be Zn, n ^ 4, or indeed have such a Zn as a generated subgraph.

If 1, 2, • • • , n is a permutation which puts R into Robinson form, one argues that the chain 1, 2, • • • , n defines a maximal spanning tree. In particular, one argues that in applying the greedy algorithm, it is always best to add an edge of the form {/, / +1}. Details are left to the reader. " One often begins by finding a maximal spanning tree in G(R) and working from there. See Hubert (1974) for details. 4. Uniqueness. In seriation, it is possible to end up with more than one proper serial order.

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1/2-Transitive Graphs of Order 3p by Alspach B., Xu M.Y.

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