Maximum weight induced multicliques and complete multipartite subgraphs in directed path overlap graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550061
Author(s):  
Fanica Gavril
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Connected Components ◽  
Directed Path ◽  
Maximum Weight ◽  
Directed Tree ◽  
Directed Paths ◽  
Multipartite Graph ◽  
Overlap Graphs ◽  
Complete Multipartite

A graph is a directed path overlap graph if it is the overlap graph of family of directed paths in a rooted directed tree. A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if it is the complement of a multiclique. A graph is a multiclique-multipartite graph if its vertex set has a partition [Formula: see text], [Formula: see text] such that [Formula: see text] is complete multipartite, [Formula: see text] is a multiclique and every two vertices [Formula: see text], [Formula: see text] are adjacent. We describe a polynomial time algorithm to find a maximum weight induced complete multipartite (MWICM) subgraph in a directed path overlap graph. In addition, we describe polynomial time algorithms to find maximum weight induced (restricted) multicliques (MWIM) and multiclique-multipartite (MWIMM) subgraphs in directed path overlap graphs.

scholarly journals Extension of Gyárfás-Sumner Conjecture to Digraphs

10.37236/9906 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Pierre Aboulker ◽  
Pierre Charbit ◽  
Reza Naserasr
Keyword(s):  
Chromatic Number ◽  
Directed Path ◽  
Complete Multipartite Graph ◽  
Color Class ◽  
Acyclic Digraph ◽  
Minimum Number ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Dichromatic Number

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices  in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization  of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph.  Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

ALGORITHMS ON SUBGRAPH OVERLAP GRAPHS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450018 ◽  
Author(s):  
FANICA GAVRIL
Keyword(s):  
Polynomial Time ◽  
Intersection Graph ◽  
Maximum Weight ◽  
Intersection Graphs ◽  
Polynomial Time Algorithms ◽  
The Family ◽  
Overlap Graphs ◽  
Hereditary Properties

Consider a graph D and a family FI of connected edge subgraphs of D. Let GI(V, F) be the intersection graph of FI and G the overlap graph of FI. We describe polynomial time algorithms for subgraph overlap graphs G when their intersection graphs GI have specific hereditary properties. The algorithms are to find maximum induced complete bipartite subgraphs, maximum weight holes of a given parity, minimum dominating holes, antiholes of a given parity and some others. In addition, we define the family of subgraph filament graphs based on D, FI and GI, and prove it to be the same as the family of subgraph overlap graphs.

scholarly journals Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

2019 ◽  
Vol 30 (02) ◽  
pp. 197-230 ◽  
Author(s):  
Markus Chimani ◽  
Giuseppe Di Battista ◽  
Fabrizio Frati ◽  
Karsten Klein
Keyword(s):  
Polynomial Time ◽  
Spanning Trees ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Negative Instance ◽  
Discrete Mathematics ◽  
Connected Components ◽  
Linear Time Algorithm ◽  
Siam Journal

In this paper, we show a polynomial-time algorithm for testing [Formula: see text]-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph [Formula: see text] with [Formula: see text] connected components [Formula: see text], do spanning trees of [Formula: see text] exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if [Formula: see text]; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which [Formula: see text] is a [Formula: see text]-planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph [Formula: see text], an instance [Formula: see text] of the pssttm problem can be constructed in polynomial time such that [Formula: see text] is [Formula: see text]-planar if and only if [Formula: see text] admits a solution. We show that, if [Formula: see text] has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance [Formula: see text] of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on [Formula: see text] is to repeatedly perform a sequence of tests, which might let us conclude that [Formula: see text] is a negative instance, and simplifications, which might let us simplify [Formula: see text] by removing or contracting some edges. Most of these tests and simplifications are performed “locally”, by looking at the crossings involving a single edge or face of a connected component [Formula: see text] of [Formula: see text]; however, some tests and simplifications have to consider certain global structures in [Formula: see text], which we call [Formula: see text]-donuts. If no test concludes that [Formula: see text] is a negative instance of the pssttm problem, then the simplifications eventually transform [Formula: see text] into an equivalent [Formula: see text]-planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Indirect identification of horizontal gene transfer

2021 ◽  
Vol 83 (1) ◽  
Author(s):  
David Schaller ◽  
Manuel Lafond ◽  
Peter F. Stadler ◽  
Nicolas Wieseke ◽  
Marc Hellmuth
Keyword(s):  
Gene Transfer ◽  
Horizontal Gene Transfer ◽  
Polynomial Time ◽  
Divergence Time ◽  
Recognition Algorithm ◽  
Vertex Coloring ◽  
Implicit Methods ◽  
Colored Graphs ◽  
Wide Range ◽  
Complete Multipartite

AbstractSeveral implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

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