Covering 2‐connected 3‐regular graphs with disjoint paths

Yu, Gexin

doi:10.1002/jgt.22219

Cited by 13 publications

(18 citation statements)

References 11 publications

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“…This conjecture might seem far-fetched, given that Reed [28] claims that there exist 2-connnected cubic graphs G with µ(G) ≥ ⌈ n 10 ⌉ and by Claim 1.3 ml(G) ≥ µ(G) + 1 always holds. However, Reed's example only gives graphs with µ(G) ≥ ⌈ n 20 ⌉, as pointed out by Yu [35]. Yu also gives 2-connnected cubic graphs G with µ(G) ≥ ⌈ n 14 ⌉ in Theorem 1.2 of [35] and this is the best known bound to date (later we prove that Yu's graphs have path covering number exactly n 14 ).…”

Section: General Casementioning

confidence: 70%

“…However, Reed's example only gives graphs with µ(G) ≥ ⌈ n 20 ⌉, as pointed out by Yu [35]. Yu also gives 2-connnected cubic graphs G with µ(G) ≥ ⌈ n 14 ⌉ in Theorem 1.2 of [35] and this is the best known bound to date (later we prove that Yu's graphs have path covering number exactly n 14 ). In order to prove that Conjecture 3.1 is sharp, let G be the graph obtained from a cycle of length k by substituting all vertices of the cycle by an edgedeleted Petersen graph P ′ (that is, P ′ is a graph obtained from the Petersen graph by deleting an edge), such that the edges of the cycle are connected to the vertices that are incident with the deleted edge of the respective copy of P ′ (see Figure 1).…”

Section: General Casementioning

confidence: 70%

“…If G is a connected cubic graph of order n then ml(G) ≤ n 6 + 1 3 . In order to prove Theorem 3.1 let us choose a minimal vertex disjoint path (vdp) cover S of G. By the theorem of Yu [35], S has size at most n 10 . A path P is said to be long, if it contains at least 18 vertices, otherwise it is short.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

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On the minimum leaf number of cubic graphs

Goedgebeur

Ozeki

Cleemput

et al. 2019

Discrete Mathematics

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The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G. We present new results concerning the minimum leaf number of cubic graphs: we show that if G is a connected cubic graph of order n, then ml(G) ≤ n 6 + 1 3 , improving on the best known result in [29] and proving the conjecture in [37]. We further prove that if G is also 2-connected, then ml(G) ≤ n 6.53 , improving on the best known bound in [5]. We also present new conjectures concerning the minimum leaf number of several types of cubic graphs and examples showing that the bounds of the conjectures are best possible.

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Section: General Casementioning

confidence: 70%

Section: General Casementioning

confidence: 70%

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On the minimum leaf number of cubic graphs

Goedgebeur

Ozeki

Cleemput

et al. 2019

Discrete Mathematics

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“…The path cover number of regular graphs [15] are also discussed. Recently, Chen et al [4] gave the following result.…”

Section: Introductionmentioning

confidence: 99%

On the Path Cover Number of Connected Quasi-Claw-Free Graphs

Liu

Zhang

et al. 2021

IEEE Access

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Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P| : P is a path cover of G}. In this paper, we show that if the minimum degree sum of an independent set with k + 1 vertices in a connected quasi-claw-free graph G of order n is no less than n − k, then p(G) ≤ k − 1, where k ≥ 2. Examples illustrate that the degree sum condition in our result is sharp. INDEX TERMS Path cover number, quasi-claw-free graph, vertex disjoint path.

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“…More recently, Magnant et al [MWY16] showed some upper bounds for general graphs depending on the maximum and minimum degree of the graph. In 2017, Yu [Yu18], motivated by the conjecture of Reed [Ree96], showed an upper bound of n 10 for the minimum path cover number in cubic 2-connected graphs. Although this is the same bound conjectured by Reed, Yu showed that the example given by Reed was not correct.…”

Section: Introductionmentioning

confidence: 99%

Covering graphs by nontrivial paths

Diaz¹

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In this thesis our aim is to study problems concerning path covers of graphs. Let G be a graph and let P be a set of pairwise vertex-disjoint paths in G. We say that P is a path cover if every vertex of G belongs to a path in P. In the minimum path cover problem, one wishes to find a path cover of minimum cardinality. In this problem, known to be NP-hard, the set P may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we consider the corresponding existence problem, and show how to reduce it to a matching problem. From this reduction, we derive a characterization that allows us to find, in polynomial time, a certificate for both the yes-answer and the no-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the cover. We show that the minimization problem on feasible instances is computationally equivalent to the minimum path cover problem: their optimum values coincide and they have the same approximation threshold. Moreover, we propose integer linear formulations for these problems and present some experimental results. For the maximization problem, we show that it can be solved in polynomial time. Finally, we also consider a weighted version of the path cover problem, in which we seek for a path cover with minimum or maximum total weight (the number of paths does not matter), and we show that while the first is polynomial, the second is NP-hard. Furthermore, for the maximization case, we show a constant-factor approximation algorithm. We also show that, when the input graph has bounded treewidth, both problems can be solved in linear time. To conclude, we present an integer linear formulation for the case of minimum total weight, and study the polytope obtained when the integrality constraint is relaxed. We show that there are graphs for which this polytope has fractional vertices, and we exhibit some classes of inequalities that are valid for the integral polytope and separate these vertices.

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Covering 2‐connected 3‐regular graphs with disjoint paths

Cited by 13 publications

References 11 publications

On the minimum leaf number of cubic graphs

On the minimum leaf number of cubic graphs

On the Path Cover Number of Connected Quasi-Claw-Free Graphs

Covering graphs by nontrivial paths

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