Detecting criticalnodesinsparsegraphs

Identifying criticalnodesinagraphisimportanttounderstandthestructuralcharacteristicsandthe connectivity propertiesofthenetwork.Inthispaper,wefocusondetectingcriticalnodes,ornodes whose deletionresultsintheminimumpair-wiseconnectivityamongtheremainingnodes.Thisproblem, known asthe critical nodeproblem has applicationsinseveralfieldsincludingbiomedicine,telecom- munications, andmilitarystrategicplanning.Weshowthattherecognitionversionoftheproblemis NP-complete andderiveamathematicalformulationbasedonintegerlinearprogramming.Inaddition, we proposeaheuristicfortheproblemwhichexploitsthecombinatorialstructureofthegraph.The heuristic isthenenhancedbytheapplicationofalocalimprovementmethod.Acomputationalstudyis presented inwhichweapplytheintegerprogrammingformulationandtheheuristictorealandrandomly generated datasets.Forallinstancestested,theheuristicisabletoefficientlyprovideoptimalsolutions in afractionofthetimerequiredbyacommercialsoftwarepackage.