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A Fully Dynamic Algorithm for Modular Decomposition and Recognition Of A Fully Dynamic Algorithm for Mo dular Decomp osition and Recognition of Cographs Ron Shamir School of Computer Science TelAviv University TelAviv Israel Ro ded Sharan International Computer Science Institute Center St Suite Berkeley CA Abstract The problem of dynamically recognizing a graph prop erty calls for eciently decid ing if an input graph satises the prop erty under rep eated mo dications to its set of vertices and edges The input to the problem consists of a series of mo dications to b e p erformed on the graph The ob jective is to maintain a representation of the graph as long as the prop erty holds and to detect when it ceases to hold In this pap er we solve the dynamic recognition problem for the class of cographs and some of its sub classes Our approach is based on maintaining the mo dular decomp osition tree of the dynamic graph and using this tree for the recognition We give the rst fully dynamic algorithm for maintaining the mo dular decomp osition tree of a cograph We thereby obtain fully dynamic algorithms for the recognition of cographs threshold graphs and trivially p erfect graphs All these algorithms work in constant time per edge mo dication and O d time per ddegree vertex mo dication Key words Fully dynamic algorithm cograph recognition mo dular decomp osition Intro duction In a dynamic graph problem one has to maintain a graph representation throughout a series of online mo dications ie insertions or deletions of a Email addresses rshamirtauacil Ron Shamir rodedicsiberkeleyedu Ro ded Sharan Preprint submitted to Elsevier Science March vertex or an edge The representation should allow to answer queries regarding certain prop erties of the dynamic graph eg is it connected Algorithms for the problem are called dynamic algorithms and are categorized dep end ing on the mo dication op erations they supp ort An incremental decremen tal algorithm supp orts only vertex insertions deletions An additionsonly deletionsonly algorithm supp orts only edge additions deletions An edges only ful ly dynamic algorithm supp orts b oth edge additions and edge deletions A ful ly dynamic algorithm supp orts edge mo dications as well as vertex mo d ications This pap er investigates dynamic recognition problems in which the queries are of the form Do es the graph belong to a certain class An algorithm for the problem is required to maintain a representation of the dynamic graph as to b elong to long as it b elongs to and to detect when it ceases Several authors have studied the problem of dynamically recognizing sp e cic graph families Hell Shamir and Sharan have given a near optimal fully dynamic algorithm for recognizing prop er interval graphs which works in O d log n time per mo dication involving d edges ie d in case of an edge mo dication and d is the degree in case of a vertex mo dication Throughout we denote the number of vertices and edges in a graph by n and m resp ectively Ibarra has given an edgesonly fully dynamic algorithm for chordal graph recognition whic h handles each edge op eration in O n time and an edgesonly fully dynamic algorithm for split graph recognition which handles each edge op eration in constant time Recently Ibarra has also devised an edgesonly fully dynamic algorithm for interval graph recognition which handles each edge op eration in O n log n time Incremental recognition algorithms were given by Hsu for interval graphs and by Deng Hell and Huang for connected prop er interval graphs A very useful representation of a graph is its mo dular decomp osition tree we defer technical denitions to Section The problem of generating the mo dular decomp osition tree of a graph was studied by many authors and several lineartime algorithms were develop ed for it For the problem of dynamically maintaining the mo dular decomp osition tree of a graph only two partial results are known Muller and Spinrad have given an incremental algorithm for mo dular decomp osition which handles each vertex insertion in O n time Corneil Perl and Stewart have given an optimal incremental the recognition and mo dular decomp osition of cographs which algorithm for handles the insertion of avertex of degree d in O d time In this pap er we give the rst fully dynamic algorithm for maintaining the mo dular decomp osition tree of a cograph Our algorithm works in O dtime per op eration involving d edges Based on this algorithm we develop fully dynamic algorithms for the recognition of cographs threshold graphs and trivially p erfect graphs All these algorithms handle a mo dication involving d edges in O dtime This is optimal with resp ect to all op erations with the p ossible exception of vertex deletion The pap er is organized as follows Section contains the denitions and the terminology used in the pap er Section presents the fully dynamic algorithm for recognizing cographs and maintaining their mo dular decomp osition tree Section contains the recognition algorithms for threshold graphs and trivially p erfect graphs Preliminaries We provide here some basic denitions and background We refer the reader to for further background reading All graphs in this pap er are simple and undirected Let G V E be a graph We denote its set of edges E also by E G For a subset R V we denote by GR the subgraph induced by the vertices in R The complement of G is the graph G V E where E fu v E u v gThe complementconnectedcomponents of G are the connected comp onents of G The graph P is a path on four vertices The graph C is a cycle on four vertices Foravertex v V we denote by N v the open neighborhood of v consisting of all neighb ors of v We let N v N v fv g For a new vertex z V andasetofedges E between z and vertices of V we z denote by G z the graph V fz gE E obtained by adding z to GFor a z vertex z V we denote by G n z the graph GV nfz g obtained by removing z from G A module M in G is a set of vertices M V suchthatevery vertex in V n M is either adjacenttoevery vertex in M or nonadjacenttoevery vertex in M A mo dule M is called trivial if M V or M contains a single vertex M is called connected if GM is a connected subgraph M is called complement GM is a connected graph For brevity we shall often refer to a connected if mo dule as if it was the subgraph induced by its vertices For example we shall talk ab out the connected comp onents of a mo dule A disconnected mo dule is called paral lel A complementdisconnected mo dule is called series A mo dule which is b oth connected and complementconnected is called a neighborhood mo dule Note that every mo dule is exactly one of the three typ es Series parallel or neighborhood A mo dule M is strong if for anymoduleN with N M wehave N M or M N A strong mo dule M is a maximal submodule of a mo dule N M if no strong submo dule of N prop erly contains M and is prop erly contained in N It has b een shown cf that every vertex of a nontrivial mo dule M is in a unique maximal submo dule of M Clearly the maximal submo dules of a parallel mo dule are its connected comp onents and the maximal submo d ules of a series mo dule are its complementconnected comp onents Hence the structure of the mo dules of a graph G can b e captured by the following mod ular decomposition tree T The no des of T corresp ond to strong mo dules of G G G The ro ot no de is V and the set of leaves of T consists of all the vertices of G GThe children of every internal no de M of T are the maximal submo dules G of M Eachinternal no de in T is lab eled series parallel or neighb orho o d G dep ending on the typ e of its corresp onding mo dule Note that the mo dular decomp osition tree of a given graph is unique In the sequel we denote the mo dular decomp osition tree of a graph G by T G We refer to a no de M of T by the set of vertices it represents that is the set G of vertices in the leaves of the subtree ro oted at M For twovertices u v V we denote by M the least common ancestor of fug and fv g in T uv G Let b e a graph class A ful ly dynamic algorithm for recognition maintains a data structure of the current graph G V E and supp orts the following op erations Edge Insertion Given a nonedge u v E up date the data structure if G fu v g or output F alse and halt otherwise Edge Deletion Given an edge u v E up date the data structure if G nfu v g or output False and halt otherwise Vertex Insertion Given a new vertex v V and a set of edges b etween v and vertices of G up date the data structure if G v or output False and halt otherwise Vertex Deletion Given a vertex v V up date the data structure if G n v or output False and halt otherwise T raditionally fully dynamic algorithms handle only edge mo dications since vertex mo dications can b e p erformed by a series of edge mo dications For example in dynamic graph connectivity adding avertex of degree d is equiv alent to adding an isolated vertex and then adding its edges one by one However in our context wehave to b e more careful since wemay not b e able to add or delete one edge at a time without ceasing to satisfy prop erty and even if there is a way to do that it might be nontrivial to nd it In other but adding or words adding or deleting a vertex can preserve the prop erty removing one edge at a time might fail to do so Hence vertex mo dications must be handled separately by the dynamic algorithm A Reduction G A graph class is called complementinvariant if G implies Examples for complementinvariant classes include perfect graphs cographs split graphs threshold graphs and permutation
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