A multi-attribute ranking solutions confirmation procedure

Ranking problems arise from the knowledge of several binary relations defined on a set of alternatives, which we intend to rank. In a previous work, the authors introduced a tool to confirm the solutions of multi-attribute ranking problems as linear extensions of a weighted sum of preference relations. An extension of this technique allows the recognition of critical preference pairs of alternatives, which are often caused by inconsistencies. Herein, a confirmation procedure is introduced and applied to confirm the results obtained by a multi-attribute decision methodology on a tender for the supply of buses to the Porto Public Transport Operator. © 2005 Springer Science + Business Media, Inc.

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.