Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
On generating all maximal independent sets
Information Processing Letters
Counting clique trees and computing perfect elimination schemes in parallel
Information Processing Letters
Fixed-parameter tractability of graph modification problems for hereditary properties
Information Processing Letters
Minimal Elimination Ordering Inside a Given Chordal Graph
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
Parameterized complexity of vertex colouring
Discrete Applied Mathematics
Parameterized coloring problems on chordal graphs
Theoretical Computer Science - Parameterized and exact computation
Interval completion with few edges
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Obtaining a planar graph by vertex deletion
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
Chordal deletion is fixed-parameter tractable
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Vertex coloring of comparability+ke and -ke
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Minimal split completions of graphs
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Parameterized Complexity
Characterizing and Computing Minimal Cograph Completions
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
Characterizing and computing minimal cograph completions
Discrete Applied Mathematics
Pathwidth and searching in parameterized threshold graphs
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
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In this paper we investigate how graph problems that are NP-hard in general, but polynomially solvable on split graphs, behave on input graphs that are close to being split. For this purpose we define split+ke and split+kv graphs to be the graphs that can be made split by removing at most k edges and at most k vertices, respectively. We show that problems like treewidth and minimum fill-in are fixed parameter tractable with parameter k on split+ke graphs. Along with positive results of fixed parameter tractability of several problems on split+ke and split+kv graphs, we also show a surprising hardness result. We prove that computing the minimum fill-in of split+kv graphs is NP-hard even for k = 1. This implies that also minimum fill-in of chordal+kv graphs is NP-hard for every k. In contrast, we show that the treewidth of split+1v graphs can be computed in polynomial time. This gives probably the first graph class for which the treewidth and the minimum fill-in problems have different computational complexity.