A linear-time algorithm for finding a paired 2-disjoint path cover in the cube of a connected graph

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946–962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus.

Download Full-text

Emparelhamentos Conexos

10.5753/etc.2020.11075 ◽

2020 ◽

Author(s):

Bruno P. Masquio ◽

Paulo E. D. Pinto ◽

Jayme L. Szwarcfiter

Keyword(s):

Connected Graph ◽

Graph Matching ◽

Linear Time ◽

Time Algorithm ◽

Maximum Matching ◽

Linear Time Algorithm ◽

Induced Subgraphs ◽

Matching Problems ◽

General Graph ◽

The One

Graph matching problems are well known and studied, in which we want to find sets of pairwise non-adjacent edges. Recently, there has been an interest in the study of matchings in which the induced subgraphs by the vertices of matchings are connected or disconnected. Although these problems are related to connectivity, the two problems are probably quite different, regarding their complexity. While the complexity of finding a maximum disconnected mat- ching is still unknown for a general graph, the one for connected matchings can be solved in polynomial time. Our contribution in this paper is a linear time algorithm to find a maximum connected matching of a general connected graph, given a general maximum matching as input.

Download Full-text

A Linear Time Algorithm for the 1-Fixed-Endpoint Path Cover Problem on Interval Graphs

SIAM Journal on Discrete Mathematics ◽

10.1137/140981265 ◽

2017 ◽

Vol 31 (1) ◽

pp. 210-239 ◽

Cited By ~ 2

Author(s):

Peng Li ◽

Yaokun Wu

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Interval Graphs ◽

Linear Time Algorithm ◽

Path Cover ◽

Cover Problem

Download Full-text

Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

The ANZIAM Journal ◽

10.1017/s1446181100013894 ◽

2002 ◽

Vol 44 (2) ◽

pp. 193-204 ◽

Cited By ~ 10

Author(s):

D. S. Franzblau ◽

A. Raychaudhuri

Keyword(s):

Linear Time ◽

Optimal Path ◽

Hamiltonian Path ◽

Maximum Flow ◽

Time Algorithm ◽

Frequency Assignment ◽

Minimum Size ◽

Linear Time Algorithm ◽

Path Cover ◽

Vertex Disjoint

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

Download Full-text

Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

Download Full-text