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**1 - 3**of**3**### Fair Sets of Some Classes of Graphs

, 2013

"... Given a non empty set S of vertices of a graph, the partiality of a vertex with respect to S is the difference between maximum and minimum of the distances of the vertex to the vertices of S. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair ..."

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Given a non empty set S of vertices of a graph, the partiality of a vertex with respect to S is the difference between maximum and minimum of the distances of the vertex to the vertices of S. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of Kn, Km,n, Kn−e, wheel graphs, odd cycles and symmetric even graphs are identified. The fair sets of the Cartesian product graphs are also discussed.

### Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs *

"... Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper ..."

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Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper interval graphs and on bipartite permutation graphs. We show that ISI can be solved in polynomial time on proper interval graphs and on bipartite permutation graphs, provided that H is connected. As a consequence, we obtain that ISI is fixed-parameter tractable on these two graph classes, when parametrised by the number of connected components of H. Our results contrast and complement the following known results: W [1]-hardness of ISI on interval graphs when parametrised by the number of vertices of H, NP-completeness of ISI on connected interval graphs and on connected permutation graphs, and NP-completeness of Subgraph Isomorphism on connected proper interval graphs and connected bipartite permutation graphs.