scholarly journals Linial's Dual Conjecture for Path-Spine Digraphs

2020 ◽  
Author(s):  
Vinícius De Souza Carvalho ◽  
Cândida Nunes Da Silva ◽  
Orlando Lee
Keyword(s):  
Stable Sets ◽  
Hamilton Path ◽  
Directed Paths ◽  
Vertex Disjoint ◽  
Single Path

 Given a digraph D, a coloring 𝒞 of D is a partition of V(D) into stable sets. The k-norm of 𝒞 is defined as ΣC∈𝒞 min{|C|, k}. A coloring of D with minimum k-norm has its k-norm noted by χk(D). A (path)-k-pack of a digraph D is a set of k vertex-disjoint (directed) paths of D. The weight of a k-pack is the number of vertices covered by the k-pack. We denote by λk(D) the weight of a maximum k-pack. Linial conjectured that χk(D) ≤ λk(D) for every digraph. Such conjecture remains open, but has been proved for some classes of digraphs. We prove the conjecture for path-spine digraphs, defined as follows. A digraph D is path-spine if there exists a partition {X, Y} of V(D) such that D[X] has a Hamilton path and every arc in D[Y] belongs to a single path Q. 

The Complexity of Deciding Strictly Non-Blocking Concentration and Generalized-Concentration Properties

1997 ◽  
Vol 08 (03) ◽  
pp. 237-252 ◽  
Author(s):  
H. K. Dai
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Strict Sense ◽  
Small Depth ◽  
Directed Paths ◽  
Concentration Properties ◽  
Vertex Disjoint ◽  
Matching Techniques

Concentrators and generalized-concentrators are interconnection networks that provide respectively pairwise vertex-disjoint directed paths and trees to satisfy interconnection requests. An interconnection network is non-blocking in the strict sense if every compatible interconnection request can be satisfied by a path regardless of any existing interconnections. We present polynomial time computational complexity results for deciding the strictly non-blocking concentration and generalized-concentration properties with small depth, by using b-matching techniques.

scholarly journals On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

Algorithmica ◽  
2019 ◽  
Vol 82 (6) ◽  
pp. 1616-1639
Author(s):  
Júlio Araújo ◽  
Victor A. Campos ◽  
Ana Karolinna Maia ◽  
Ignasi Sau ◽  
Ana Silva
Keyword(s):  
Directed Paths ◽  
Vertex Disjoint

scholarly journals On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

2018 ◽  
pp. 66-79 ◽  
Author(s):  
Júlio Araújo ◽  
Victor A. Campos ◽  
Ana Karolinna Maia ◽  
Ignasi Sau ◽  
Ana Silva
Keyword(s):  
Directed Paths ◽  
Vertex Disjoint

scholarly journals Computing directed Steiner path covers

2021 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs ◽  
Jochen Rethmann ◽  
Egon Wanke
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Time Algorithm ◽  
Integer Programs ◽  
Path Cover ◽  
Hamiltonian Path Problem ◽  
Directed Paths ◽  
Minimum Number ◽  
Cover Problem ◽  
Vertex Disjoint

AbstractIn this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

Partitioning a graph into vertex-disjoint paths

2005 ◽  
Vol 42 (3) ◽  
pp. 277-294
Author(s):  
Jianping Li ◽  
George Steiner
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Disjoint Paths ◽  
Extremal Graphs ◽  
Hamilton Path ◽  
Vertex Disjoint ◽  
Positive Integers

Let G=(V,E) be a simple graph of order n. We consider the problem of partitioning G into vertex-disjoint paths. We obtain the following new results: (i) For any positive integer k, if dG(x)+dG(y) = n-k-1 for every pair x, y of nonadjacent vertices in  G, then G can be partitioned into k vertex-disjoint paths, unless G belongs to certain classes of extremal graphs which we characterize; (ii) For the case k=2, we strengthen our result by showing that for any two positive integers p1 and p2 satisfying n= p1+ p2,if dG(x)+dG(y) = n-3  for every pair x, y of nonadjacent vertices in G and G does not belong to classes of exceptional graphs we define, then G can be partitioned into two vertex-disjoint paths  P1 and P2 of order p1 and p2,  respectively. These results are generalizations of some classical results of Dirac and Ore, and also lead to new sufficient conditions for the existence of a Hamilton path in a graph.

scholarly journals Linial’s Conjecture for Arc-spine Digraphs

2019 ◽  
Vol 17 (2) ◽  
Author(s):  
Lucas Rigo Yoshimura ◽  
Maycon Sambinelli ◽  
Cândida Nunes da Silva ◽  
Orlando Lee
Keyword(s):  
Positive Integer ◽  
Stable Set ◽  
Stable Sets ◽  
Directed Paths ◽  
Path Partition

A path partition P of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as Sum (p in P) min{|p_i|, k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacentvertices of V(D). Given a positive integer k, we denote by alpha_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that pi_k(D) ≤ alpha_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial’s Conjecture for arc-spine digraphs.

scholarly journals Some Partial Results on Linial's Conjecture for Matching-Spine Digraphs

2021 ◽  
Author(s):  
Jadder Bismarck de Sousa Cruz ◽  
Cândida Nunes da Silva ◽  
Orlando Lee
Keyword(s):  
Positive Integer ◽  
Optimal Path ◽  
Disjoint Paths ◽  
Stable Sets ◽  
Maximum Weight ◽  
Hamilton Path ◽  
Vertex Set ◽  
Path Partition

Let $k$ be a positive integer. A \emph{partial $k$-coloring} of a digraph $D$ is a set $\calC$ of $k$ disjoint stable sets and has \emph{weight} defined as $\sum_{C \in \calC} |C|$. An \emph{optimal} $k$-coloring is a $k$-coloring of maximum weight. A \emph{path partition} of a digraph $D$ is a set $\calP$ of disjoint paths of $D$ that covers its vertex set and has \emph{$k$-norm} defined as $\sum_{P \in \mathcal{P}} \min\{|P|,k\}$. A path partition $\calP$ is \emph{$k$-optimal} if it has minimum $k$-norm. A digraph $D$ is \emph{matching-spine} if its vertex set can be partitioned into sets $X$ and $Y$, such that $D[X]$ has a Hamilton path and the arc set of $D[Y]$ is a matching. Linial (1981) conjectured that the $k$-norm of a $k$-optimal path partition of a digraph is at most the weight of an optimal partial $k$-coloring. We present some partial results on this conjecture for matching-spine digraphs.

scholarly journals The Proofs of Two Directed Paths Conjectures of Bollobás and Leader

2017 ◽  
Vol 26 (5) ◽  
pp. 762-774
Author(s):  
TREVOR PINTO
Keyword(s):  
Disjoint Paths ◽  
Directed Edge ◽  
Edge Disjoint ◽  
Directed Paths ◽  
Disjoint Sets ◽  
A Chain ◽  
Vertex Disjoint

Let A and B be disjoint sets, of size 2k, of vertices of Qn, the n-dimensional hypercube. In 1997, Bollobás and Leader proved that there must be (n − k)2k edge-disjoint paths between such A and B. They conjectured that when A is a down-set and B is an up-set, these paths may be chosen to be directed (that is, the vertices in the path form a chain). We use a novel type of compression argument to prove stronger versions of these conjectures, namely that the largest number of edge-disjoint paths between a down-set A and an up-set B is the same as the largest number of directed edge-disjoint paths between A and B. Bollobás and Leader made an analogous conjecture for vertex-disjoint paths, and we prove a strengthening of this by similar methods. We also prove similar results for all other sizes of A and B.

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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