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List Partitions
 Proc. 31st Ann. ACM Symp. on Theory of Computing
, 2003
"... List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, ..."
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Cited by 26 (11 self)
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List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, finding homogeneous sets, joins, clique cutsets, stable cutsets, skew cutsets and so on. We develop tools which allow us to classify the complexity of many list partition problems and, in particular, yield the complete classification for small matrices M . Along the way, we obtain a variety of specific results including: generalizations of Lov'asz's communication bound on the number of cliqueversus stableset separators; polynomialtime algorithms to recognize generalized split graphs; a polynomial algorithm for the list version of the Clique Cutset Problem; and the first subexponential algorithm for the Skew Cutset Problem of Chv'atal. We also show that the dichotomy (NP complete versus polynomialtime solvable), conjectured for certain graph homomorphism problems would, if true, imply a slightly weaker dichotomy (NP complete versus quasipolynomial) for our list partition problems 1 . Email: tomas@theory.stanford.edu. y School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A1S6. Email: pavol@cs.sfu.ca. Supported by a Research Grant from the National Sciences and Engineering Research Council. z Departamento da Ciencia da Computac~ao  I.M., COPPE/Sistemas, Universidade Federal do Rio de Janeiro, RJ, 21945970, Brasil. Email: sula@cos.ufrj.br. Supported by CNPq and PRONEX 107/97. x Department of Computer Science, Stanford University, CA 943059045. Email: rajeev@cs.stanford.edu. Supported by an ARO MURI Grant DAAH04961...
Partitioning chordal graphs into independent sets and cliques
 Discrete Applied Math
"... We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k = l = 1). Much of the appeal of split graphs is due to the fact that they are chordal, a p ..."
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Cited by 19 (6 self)
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We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k = l = 1). Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)graphs in general. (For instance, being a (k,0)graph is equivalent to being kcolourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)graphs. We also give an O(n(m + n)) recognition algorithm for chordal (k,ℓ)graphs. When k = 1, our algorithm runs in time O(m + n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list ’ (or ‘precolouring extension’) version of the split partition problem given a graph with some vertices preassigned to the independent set, or to the clique, is there a split partition extending this preassignment Preprint submitted to Elsevier Science 13 December 2003 Another way to think of our main result is the following minmax property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr’s equals the minumum number of cliques that meet all Kr’s. Key words:
VertexPartitioning into Fixed Additive InducedHereditary Properties Is NPhard
 J. Combin. Cit
, 2004
"... Can the vertices of an arbitrary graph G be partitioned into A∪B, so that G[A] is a linegraph and G[B] is a forest? Can G be partitioned into a planar graph and a perfect graph? The NPcompleteness of these problems are special cases of our result: if P and Q are additive inducedhereditary gra ..."
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Cited by 12 (5 self)
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Can the vertices of an arbitrary graph G be partitioned into A∪B, so that G[A] is a linegraph and G[B] is a forest? Can G be partitioned into a planar graph and a perfect graph? The NPcompleteness of these problems are special cases of our result: if P and Q are additive inducedhereditary graph properties, then (P,Q)colouring is NPhard, with the sole exception of graph 2colouring (the case where both P and Q are the Set O of finite edgeless graphs). Moreover, (P,Q)colouring is NPcomplete iff P and Qrecognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many subcases.
RECOGNIZING PERFECT 2SPLIT GRAPHS
, 2000
"... A graph is a split graph if its vertices can be partitioned into a clique and a stable set. A graph is a ksplit graph if its vertices can be partitioned into k sets, each of which induces a split graph. We show that the strong perfect graph conjecture is true for 2split graphs and we design a poly ..."
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Cited by 4 (0 self)
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A graph is a split graph if its vertices can be partitioned into a clique and a stable set. A graph is a ksplit graph if its vertices can be partitioned into k sets, each of which induces a split graph. We show that the strong perfect graph conjecture is true for 2split graphs and we design a polynomial algorithm to recognize a perfect 2split graph.
Uniqueness and Complexity in Generalised Colouring
, 2003
"... The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In th ..."
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Cited by 3 (2 self)
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The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced...
Partitions of graphs into trees
 IN PROCEEDINGS OF GRAPH DRAWING’06 (KARLSRUHE), VOLUME 4372 OF LNCS
, 2007
"... In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) ..."
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Cited by 3 (0 self)
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In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3tree partition of the inner edges. Maximal planar bipartite graphs have a 2tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NPhardness of the ktree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].
Bisplit Graphs
 DIMACS TR 200244. T. FEDER ET AL. / THEORETICAL COMPUTER SCIENCE 349 (2005) 52 – 66
, 2002
"... A graph is bisplit if it can be partitioned into a stable set and a biclique (i.e. a complete bipartite graph). We provide an O(nm) recognition algorithm for these graphs, and characterize them in terms of forbidden induced subgraphs. Moreover, ..."
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Cited by 2 (0 self)
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A graph is bisplit if it can be partitioned into a stable set and a biclique (i.e. a complete bipartite graph). We provide an O(nm) recognition algorithm for these graphs, and characterize them in terms of forbidden induced subgraphs. Moreover,
On the Complexity of (k, l)Graph Sandwich Problems
"... A graph G is (k, l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k, l)Graph Sandwich Problem asks, given two graphs G¹ = (V, E¹) and G² = (V, E²), whether there exists a graph G = (V, E) such that E¹ ⊆ E ⊆ E² and G ..."
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A graph G is (k, l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k, l)Graph Sandwich Problem asks, given two graphs G¹ = (V, E¹) and G² = (V, E²), whether there exists a graph G = (V, E) such that E¹ ⊆ E ⊆ E² and G is (k, l). In this paper, we prove that the (k, l)Graph Sandwich Problem is NP complete for the cases k = 1 and l = 2, k = 2 and l = 1, or k = l = 2. This completely classifies the complexity of the (k, l)Graph Sandwich Problem as follows: the problem is NPcomplete, if k + l > 2; the problem is polynomial otherwise. In addition, we consider the degree Δ constraint subproblem and completely classifies the problem as follows: the problem is polynomial, for k ≤ 2 or Δ ≤ 3; the problem is NPcomplete otherwise.