716 research outputs found
A Linear-Time Recognition Algorithm for P4-Reducible Graphs
The P4-reducible graphs are a natural generalization of the well-known class of cographs, with applications to scheduling, computational semantics, and clustering. More precisely, the P4-reducible graphs are exactly the graphs none of whose vertices belong to more than one chordless path with three edges. A remarkable property of P4-reducible graphs is their unique tree representation up to isomorphism. In this paper we present a linear-time algorithm to recognize P4-reducible graphs and to construct their corresponding tree representation
Linear Time Optimization Algorithms for P4-Sparse Graphs
Quite often, real-life applications suggest the study of graphs that feature some local density properties. In particular, graphs that are unlikely to have more than a few chordless paths of length three appear in a number of contexts. A graph G is P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize both the class of cographs and the class of P4-reducible graphs. It has been shown that P4-sparse graphs can be recognized in time linear in the size of the graph. The main contribution of this paper is to show that once the data structures returned by the recognition algorithm are in place, a number of NP-hard problems on general graphs can be solved in linear time for P4-sparse graphs. Specifically with an n-vertex P4-sparse graph as input the problems of finding a maximum size clique, maximum size stable set, a minimum coloring, a minimum covering by clique, and the size of the minimum fill-in can be solved in O(n) time, independent of the number of edges in the graph
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
FPT algorithms to recognize well covered graphs
Given a graph , let and be the sizes of a minimum and a
maximum minimal vertex covers of , respectively. We say that is well
covered if (that is, all minimal vertex covers have the same
size). Determining if a graph is well covered is a coNP-complete problem. In
this paper, we obtain -time and -time
algorithms to decide well coveredness, improving results of Boria et. al.
(2015). Moreover, using crown decomposition, we show that such problems admit
kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved
that recognizing well covered graphs is coW[2]-hard when the independence
number is the parameter. Contrasting with such
coW[2]-hardness, we present an FPT algorithm to decide well coveredness when
and the degeneracy of the input graph are aggregate parameters.
Finally, we use the primeval decomposition technique to obtain a linear time
algorithm for extended -laden graphs and -graphs, which is FPT
parameterized by , improving results of Klein et al (2013).Comment: 15 pages, 2 figure
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
On the p-Connectedness of Graphs – a Survey
A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs.
This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions
A Fast Parallel Algorithm to Recognize P4-Sparse Graphs
A number of problems in mobile computing, group-based collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain “local density” characteristics. Typically, the notion of local density is equated with the absence of chordless paths of length three or more. Recently, a new metric for local density has been proposed, allowing a number of such induced paths to occur. More precisely, a graphG is called P4-sparse if no set of five vertices inG induces more than one chordless path of length three. P4-sparse graphs generalize the well-known class of cographs corresponding to a more stringent local density metric. One remarkable feature of P4-sparse graphs is that they admit a tree representation unique up to isomorphism. In this work we present a parallel algorithm to recognize P4-sparse graphs and show how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation. With a graphG= (V, E) with¦V¦=n and¦E¦= m as input, our algorithms run inO(log n) time usingO((n2 + mn)/ log n) processors in the EREW-PRAM model
On a unique tree representation for P4-extendible graphs
AbstractSeveral practical applications in computer science and computational linguistics suggest the study of graphs that are unlikely to have more than a few induced paths of length three. These applications have motivated the notion of a cograph, defined by the very strong restriction that no vertex may belong to an induced path of length three. The class of P4-extendible graphs that we introduce in this paper relaxes this restriction, and in fact properly contains the class of cographs, while still featuring the remarkable property of admitting a unique tree representation. Just as in the case of cographs, the class of P4-extendible graphs finds applications to clustering, scheduling, and memory management in a computer system. We give several characterizations for P4-extendible graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-extendible graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time
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