Computing roots of graphs is hard
Discrete Applied Mathematics
Algorithms for Square Roots of Graphs
SIAM Journal on Discrete Mathematics
Journal of Algorithms
Graph classes: a survey
On Graph Powers for Leaf-Labeled Trees
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
On k- Versus (k + 1)-Leaf Powers
COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
Polynomial kernels for 3-leaf power graph modification problems
Discrete Applied Mathematics
Discovering pairwise compatibility graphs
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
A 2k Kernel for the cluster editing problem
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
A 2k kernel for the cluster editing problem
Journal of Computer and System Sciences
5-th phylogenetic root construction for strictly chordal graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Fixed-parameter tractable generalizations of cluster editing
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
Extending the tractability border for closest leaf powers
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Error compensation in leaf root problems
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Computing bounded-degree phylogenetic roots of disconnected graphs
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
A more effective linear kernelization for Cluster Editing
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
On the recognition of k-equistable graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
On pairwise compatibility graphs having Dilworth number two
Theoretical Computer Science
A Survey of Parallel and Distributed Algorithms for the Steiner Tree Problem
International Journal of Parallel Programming
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Given a graph G = (V, E) and a positive integer k, the PHYLOGENETIC k-ROOT PROBLEM asks for a (unrooted) tree T without degree-2 nodes such that its leaves are labeled by V and (u, v) ∈ E if and only if dT (u, v) ≤ k. If the vertices in V are also allowed to be internal nodes in T, then we have the Steiner k-ROOT PROBLEM. Moreover, if a particular subset S of V are required to be internal nodes in T, then we have the RESTRICTED STEINER k-ROOT PROBLEM. Phylogenetic k-roots and Steiner k-roots extend the standard notion of GRAPH ROOTS and are motivated by applications in computational biology. In this paper, we first present O(n + e)-time algorithms to determine if a (not necessarily connected) graph G = (V, E) has an S-restricted 1-root Steiner tree for a given subset S ⊂ V , and to determine if a connected graph G = (V, E) has an S-restricted 2-root Steiner tree for a given subset S ⊂ V, where n = |V| and e = |E|. We then use these two algorithms as subroutines to design O(n + e)-time algorithms to determine if a given (not necessarily connected) graph G = (V, E) has a 3-root phylogeny and to determine if a given connected graph G = (V, E) has a 4-root phylogeny.