On the Vertex -Path Cover

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Dec 5, 2013 - Abstract. A subset of vertices of a graph is called a -path vertex cover if every path of order in contains at least one vertexย ...
On the Vertex ๐‘˜-Path Cover Boห‡stjan Breห‡sarโˆ—, Marko Jakovacโˆ— Faculty of Natural Science and Mathematics, University of Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

Jยดan Katreniห‡cโ€ and Gabriel Semaniห‡sinโ€ก

ห‡ arik University, Koห‡sice, Slovakia Institute of Computer Science, P.J. Safยด

Andrej Taranenko

Faculty of Natural Science and Mathematics, University of Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

December 5, 2013

Abstract A subset ๐‘† of vertices of a graph ๐บ is called a ๐‘˜-path vertex cover if every path of order ๐‘˜ in ๐บ contains at least one vertex from ๐‘†. Denote by ๐œ“๐‘˜ (๐บ) the minimum cardinality of a ๐‘˜-path vertex cover in ๐บ. In this paper present an upper bound for ๐œ“3 of graphs with given average degree. We also give a lower bound for ๐œ“๐‘˜ of regular graphs. For the Cartesian products of two paths we give an asymptotically tight bound for ๐œ“๐‘˜ and the exact value for ๐œ“3 .

1

Introduction

Let ๐บ be a graph and ๐‘˜ be a positive integer. Then ๐‘† โŠ† ๐‘‰ (๐บ) is the vertex ๐‘˜-path cover of ๐บ if every path on ๐‘˜ vertices in ๐บ contains a vertex from ๐‘†. We denote by ๐œ“๐‘˜ (๐บ) the cardinality of a minimum vertex ๐‘˜-path cover in ๐บ. This graph invariant was recently introduced in [4], where the motivation for this problem arises in ensuring data integrity communication in wireless sensor networks using the ๐‘˜-generalized Canvas scheme [13]. โˆ—

Supported by the Ministry of Science and Higher Education of Slovenia under the grants J1-2043 and P1-0297. โ€  The research of the author supported in part by Slovak APVV grant SK-SI-0010-08 and Slovak VEGA grant 1/0035/09. โ€ก The research of the author supported in part by Slovak APVV grant SK-SI-0010-08, APVV-0007-07 and by the Agency of the Slovak Ministry of Education for the Structural Funds of the EU, under project ITMS:26220120007.

1

The vertex ๐‘˜-path cover can be regarded as the generalization of the vertex cover. Indeed, ๐œ“2 (๐บ) equals to the size of minimum vertex cover, and ๐œ“2 (๐บ) = โˆฃ๐‘‰ (๐บ)โˆฃ โˆ’ ๐›ผ(๐บ) where ๐›ผ(๐บ) stands for the independence number. The value of ๐œ“3 (๐บ) corresponds to the concept of dissociation number of a graph [17], de๏ฌned as follows. A subset of vertices in a graph ๐บ is called dissociation if it induces a subgraph with maximum degree 1. The number of vertices in a maximum cardinality dissociation set in ๐บ is called the dissociation number of ๐บ and is denoted ๐‘‘๐‘–๐‘ ๐‘ (๐บ). Clearly ๐œ“3 (๐บ) = โˆฃ๐‘‰ (๐บ)โˆฃ โˆ’ ๐‘‘๐‘–๐‘ ๐‘ (๐บ). The problem of computing ๐‘‘๐‘–๐‘ ๐‘ (๐บ) has been introduced by Yannakakis [17], who also proved it to be NP-hard in the class of bipartite graphs. The dissociation number problem was also studied in [1, 2, 6, 8, 14, 12], see [14] also for a survey. Recently, Tu and Zhou [15] presented a 2-approximation for 3-path vertex cover problem even for the weighted case of the problem. An exact algorithm for computing ๐œ“3 (๐บ) in running time ๐‘‚(1.5171๐‘› ) for a graph of order ๐‘› was presented in [10]. The problem of computing ๐œ“๐‘˜ (๐บ) is NP-hard for each ๐‘˜ โ‰ฅ 2, but polynomial for instance in trees, as shown in [4]. The authors investigate upper bounds on the value of ๐œ“๐‘˜ (๐บ) and provide several estimations and exact values of ๐œ“๐‘˜ (๐บ). It is also proven that ๐œ“3 (๐บ) โ‰ค (2๐‘› + ๐‘š)/6, for every graph ๐บ with ๐‘› vertices and ๐‘š edges. In this paper we present a more general result showing that for an arbitrary graph ๐บ with ๐‘› vertices and ๐‘š edges and an integer ๐‘˜ such that ๐‘˜๐‘› ๐‘š ๐‘˜โ‰ค๐‘š ๐‘› โ‰ค ๐‘˜ + 1 we have ๐œ“3 (๐บ) โ‰ค ๐‘˜+2 + (๐‘˜+1)(๐‘˜+2) . In Section 3 we consider bounds on ๐‘‘-regular graphs and show that for an arbitrary integer ๐‘˜ โ‰ฅ 2 and ๐‘‘โˆ’๐‘˜+2 ๐‘‘-regular graph ๐บ, ๐‘‘ โ‰ฅ ๐‘˜ โˆ’ 1 we have ๐œ“๐‘˜ (๐บ) โ‰ฅ 2๐‘‘โˆ’๐‘˜+2 โˆฃ๐‘‰ (๐บ)โˆฃ. Section 4 is devoted to the vertex ๐‘˜-path cover number of Cartesian products of graphs, in particular on grids. We conclude this section with the following straightforward values of ๐œ“๐‘˜ . Proposition 1.1. Let ๐‘˜ โ‰ฅ 2 and ๐‘› โ‰ฅ ๐‘˜ be positive integers. Then โŒŠ โŒ‹ โˆ™ ๐œ“๐‘˜ (๐‘ƒ๐‘› ) = ๐‘›๐‘˜ , โˆ™ ๐œ“๐‘˜ (๐ถ๐‘› ) = โŒˆ ๐‘›๐‘˜ โŒ‰, โˆ™ ๐œ“๐‘˜ (๐พ๐‘› ) = ๐‘› โˆ’ ๐‘˜ + 1.

2

Upper bounds on degree of vertices

First, recall the result of [4] which is a consequence of Lovยดaszโ€™s decomposition [11] of a graph with maximum degree ฮ” into subgraphs of maximum degree 1. 2

Lemma 2.1 ([4]). Let ๐บ be a graph of maximum degree ฮ”. Then ๐œ“3 (๐บ) โ‰ค

โŒˆ ฮ”โˆ’1 2 โŒ‰ โŒˆ ฮ”+1 2 โŒ‰

โˆฃ๐‘‰ (๐บ)โˆฃ.

In the following theorem we give a tight upper bound in terms of average degree of graph, which extends the result ๐œ“3 (๐บ) โ‰ค (2๐‘› + ๐‘š)/6 of [4]. Theorem 2.1. Let ๐บ be a graph of order ๐‘›, size ๐‘š and average degree ๐‘‘(๐บ), and ๐‘˜ be a positive integer such that 2๐‘˜ โ‰ค ๐‘‘(๐บ) โ‰ค 2๐‘˜ + 2. Then ๐œ“3 (๐บ) โ‰ค

๐‘˜๐‘› ๐‘š + ๐‘˜ + 2 (๐‘˜ + 1)(๐‘˜ + 2)

Proof. Proof by induction on ๐‘˜ (the basis of induction ๐‘˜ = 1 coincides with the bound from [4]). Assume (๐‘˜ โˆ’ 1)๐‘› ๐‘š ๐œ“3 (๐บ) โ‰ค + ๐‘˜+1 ๐‘˜(๐‘˜ + 1) for all graphs with average degree ๐‘‘, 2๐‘˜ โˆ’ 2 โ‰ค ๐‘‘ โ‰ค 2๐‘˜. We aim to prove that for a graph ๐บ with average degree ๐‘‘, 2๐‘˜ โ‰ค ๐‘‘ โ‰ค 2๐‘˜ + 2 the following holds ๐‘˜๐‘› ๐‘š ๐œ“3 (๐บ) โ‰ค + . ๐‘˜ + 2 (๐‘˜ + 1)(๐‘˜ + 2) Systematically remove from ๐บ vertices of degree at least 2(๐‘˜ + 1). We get a graph ๐บโ€ฒ of ฮ”(๐บโ€ฒ ) โ‰ค 2(๐‘˜ + 1) โˆ’ 1. For ๐บโ€ฒ we now look at two cases. In the ๏ฌrst case we do not use the induction step. If ๐‘˜๐‘›โ€ฒ โ‰ค ๐‘šโ€ฒ , then by Lemma 2.1 ๐œ“3 (๐บโ€ฒ ) โ‰ค = = โ‰ค

๐‘˜ ๐‘›โ€ฒ ๐‘˜+1 ๐‘˜(๐‘˜ + 2) ๐‘›โ€ฒ (๐‘˜ + 1)(๐‘˜ + 2) ๐‘˜(๐‘˜ + 1) + ๐‘˜ โ€ฒ ๐‘› (๐‘˜ + 1)(๐‘˜ + 2) ๐‘˜ ๐‘šโ€ฒ ๐‘›โ€ฒ + ๐‘˜+2 (๐‘˜ + 1)(๐‘˜ + 2)

In the second case, if ๐‘˜๐‘›โ€ฒ โ‰ฅ ๐‘šโ€ฒ we use the induction step. ๐œ“3 (๐บโ€ฒ ) โ‰ค = โ‰ค =

๐‘˜โˆ’1 โ€ฒ ๐‘› + ๐‘˜+1 ๐‘˜โˆ’1 โ€ฒ ๐‘› + ๐‘˜+1 ๐‘˜โˆ’1 โ€ฒ ๐‘› + ๐‘˜+1 ๐‘˜ ๐‘›โ€ฒ + ๐‘˜+2

๐‘šโ€ฒ ๐‘˜(๐‘˜ + 1) 2๐‘šโ€ฒ ๐‘šโ€ฒ + ๐‘˜(๐‘˜ + 1)(๐‘˜ + 2) (๐‘˜ + 1)(๐‘˜ + 2) 2๐‘›โ€ฒ ๐‘šโ€ฒ + (๐‘˜ + 1)(๐‘˜ + 2) (๐‘˜ + 1)(๐‘˜ + 2) ๐‘šโ€ฒ (๐‘˜ + 1)(๐‘˜ + 2) 3

Next we prove a lower bound for ๐œ“3 (๐บ), showing that the upper bound of Theorem 2.1 is tight. Theorem 2.2. For every positive rational number ๐‘Ž๐‘ > 1, 2๐‘˜ โ‰ค there exists a graph ๐บ with average degree ๐‘‘(๐บ) = ๐‘Ž๐‘ and ๐œ“3 (๐บ) โ‰ฅ

๐‘Ž ๐‘

โ‰ค 2๐‘˜ + 2,

๐‘˜๐‘› ๐‘š + . ๐‘˜ + 2 (๐‘˜ + 1)(๐‘˜ + 2)

Proof. Denote by ๐ป๐‘› a complete graph on ๐‘› vertices minus one perfect matching (assume ๐‘› is even). Clearly โˆฃ๐‘‰ (๐ป๐‘› )โˆฃ = ๐‘› and โˆฃ๐ธ(๐ป๐‘› )โˆฃ =

๐‘›(๐‘› โˆ’ 1) ๐‘› ๐‘›(๐‘› โˆ’ 2) โˆ’ = . 2 2 2

Clearly ๐‘‘๐‘–๐‘ ๐‘ (๐ป๐‘› ) โ‰ค 2, because arbitrary three vertices of ๐ป๐‘› forms a path of order three. Therefore ๐œ“3 (๐ป๐‘› ) = ๐‘› โˆ’ 2, for ๐‘› โ‰ฅ 2. We construct the graph ๐บ as the disjoint union of โˆ™ ๐‘ฅ components ๐ป2๐‘˜+2 , โˆ™ ๐‘ฆ components ๐ป2๐‘˜+4 . We let ๐‘ฅ = (2๐‘ โˆ’ ๐‘Ž + 2๐‘˜๐‘)(๐‘˜ + 2) and ๐‘ฆ = (๐‘Ž โˆ’ 2๐‘˜๐‘)(๐‘˜ + 1). First, we verify the average degree of the graph ๐บ. There are ๐‘ฅ(2๐‘˜ + 2) vertices of degree 2๐‘˜ and ๐‘ฆ(2๐‘˜ + 4) vertices of degree 2๐‘˜ + 2, hence we get ๐‘‘(๐บ) = = = = =

๐‘ฅ(2๐‘˜ + 2)(2๐‘˜) + ๐‘ฆ(2๐‘˜ + 4)(2๐‘˜ + 2) ๐‘ฅ(2๐‘˜ + 2) + ๐‘ฆ(2๐‘˜ + 4) (2๐‘ โˆ’ ๐‘Ž + 2๐‘˜๐‘)(๐‘˜ + 2)(2๐‘˜ + 2)(2๐‘˜) + (๐‘Ž โˆ’ 2๐‘˜๐‘)(๐‘˜ + 1)(2๐‘˜ + 4)(2๐‘˜ + 2) (2๐‘ โˆ’ ๐‘Ž + 2๐‘˜๐‘)(๐‘˜ + 2)(2๐‘˜ + 2) + (๐‘Ž โˆ’ 2๐‘˜๐‘)(๐‘˜ + 1)(2๐‘˜ + 4) (2๐‘ โˆ’ ๐‘Ž + 2๐‘˜๐‘)(2๐‘˜) + (๐‘Ž โˆ’ 2๐‘˜๐‘)(2๐‘˜ + 2) (2๐‘ โˆ’ ๐‘Ž + 2๐‘˜๐‘) + (๐‘Ž โˆ’ 2๐‘˜๐‘) 2๐‘˜๐‘ โˆ’ (๐‘Ž + 2๐‘˜๐‘)(2๐‘˜) + (๐‘Ž โˆ’ 2๐‘˜๐‘)(2๐‘˜) + 2(๐‘Ž โˆ’ 2๐‘˜๐‘) 2๐‘ ๐‘Ž . ๐‘

Now, let us verify the value of ๐œ“3 of ๐บ. Clearly, ๐œ“3 (๐บ) = ๐‘ฅ โ‹… ๐œ“3 (๐ป2๐‘˜+2 ) + ๐‘ฆ โ‹… ๐œ“3 (๐ป2๐‘˜+4 ) = ๐‘ฅ(2๐‘˜) + ๐‘ฆ(2๐‘˜ + 2).

4

And this exactly matches the value of the formula ๐œ“3 (๐บ) = = = = =

3

๐‘˜๐‘› ๐‘š + ๐‘˜ + 2 (๐‘˜ + 1)(๐‘˜ + 2) ๐‘˜(๐‘ฅ(2๐‘˜ + 2) + ๐‘ฆ(2๐‘˜ + 4)) ๐‘ฅ(2๐‘˜)(2๐‘˜ + 2) + ๐‘ฆ(2๐‘˜ + 2)(2๐‘˜ + 4) + ๐‘˜+2 2(๐‘˜ + 1)(๐‘˜ + 2) ๐‘˜๐‘ฅ(2๐‘˜ + 2) ๐‘ฅ(2๐‘˜) + ๐‘ฆ(2๐‘˜ + 4) + 2๐‘˜๐‘ฆ + ๐‘˜+2 (๐‘˜ + 2) ๐‘˜๐‘ฅ(2๐‘˜ + 2) ๐‘ฅ(2๐‘˜) + 2๐‘˜๐‘ฆ + + 2๐‘ฆ ๐‘˜+2 (๐‘˜ + 2) 2๐‘˜๐‘ฅ + 2๐‘˜๐‘ฆ + 2๐‘ฆ.

Regular graphs

In the main theorem of this section we shall use the following result of Erdหos and Gallai [7]. Theorem 3.1 ([7]). If ๐บ is a graph on ๐‘› vertices that does not contain a path of order ๐‘˜ then it cannot have more than ๐‘›(๐‘˜โˆ’2) edges. Moreover, the 2 bound is achieved when the graph consists of disjoint complete graphs on ๐‘˜ โˆ’ 1 vertices. Theorem 3.2. Let ๐‘˜ โ‰ฅ 2 and ๐‘‘ โ‰ฅ ๐‘˜ โˆ’ 1 be positive integers. Then for any ๐‘‘-regular graph ๐บ the following holds: ๐œ“๐‘˜ (๐บ) โ‰ฅ

๐‘‘โˆ’๐‘˜+2 โˆฃ๐‘‰ (๐บ)โˆฃ. 2๐‘‘ โˆ’ ๐‘˜ + 2

Proof. Let ๐‘† โŠ† ๐‘‰ (๐บ) be a vertex ๐‘˜-path cover and ๐‘‡ = ๐‘‰ (๐บ) โˆ– ๐‘†. Let ๐ธ๐‘† , ๐ธ๐‘‡ be the number of edges with both endvertices in ๐‘† and ๐‘‡ respectively. Let ๐ธ๐‘†๐‘‡ be the number of edges with one endvertex in ๐‘† and the second vertex in ๐‘‡ . Then obviously โˆฃ๐ธ(๐บ)โˆฃ = 21 ๐‘‘โˆฃ๐‘‰ (๐บ)โˆฃ = โˆฃ๐ธ๐‘† โˆฃ + โˆฃ๐ธ๐‘†๐‘‡ โˆฃ + โˆฃ๐ธ๐‘‡ โˆฃ. Since ๐บ is ๐‘‘-regular, ๐‘‘โˆฃ๐‘†โˆฃ = 2โˆฃ๐ธ๐‘† โˆฃ + โˆฃ๐ธ๐‘†๐‘‡ โˆฃ. Therefore โˆฃ๐‘†โˆฃ โ‰ฅ ๐‘‘1 โˆฃ๐ธ๐‘†๐‘‡ โˆฃ. Similarly, โˆฃ๐ธ๐‘†๐‘‡ โˆฃ + 2โˆฃ๐ธ๐‘‡ โˆฃ = ๐‘‘โˆฃ๐‘‡ โˆฃ. Since, ๐บ[๐ธ๐‘‡ ] does not contain the path of order ๐‘˜, according to Theorem 3.1, we have โˆฃ๐ธ๐‘‡ โˆฃ โ‰ค โˆฃ๐‘‡ โˆฃ(๐‘˜โˆ’2) . Combining all 2 the previous formulas we immediately have โˆฃ๐‘†โˆฃ โ‰ฅ

1 1 1 ๐‘‘โˆ’๐‘˜+2 โˆฃ๐ธ๐‘†๐‘‡ โˆฃ = (๐‘‘โˆฃ๐‘‡ โˆฃ โˆ’ 2โˆฃ๐ธ๐‘‡ โˆฃ) โ‰ฅ (๐‘‘โˆฃ๐‘‡ โˆฃ โˆ’ โˆฃ๐‘‡ โˆฃ(๐‘˜ โˆ’ 2)) = โˆฃ๐‘‡ โˆฃ. ๐‘‘ ๐‘‘ ๐‘‘ ๐‘‘

Then โˆฃ๐‘†โˆฃ + โˆฃ๐‘‡ โˆฃ โˆฃ๐‘‡ โˆฃ ๐‘‘ 2๐‘‘ โˆ’ ๐‘˜ + 2 =1+ โ‰ค1+ = โˆฃ๐‘†โˆฃ โˆฃ๐‘†โˆฃ ๐‘‘โˆ’๐‘˜+2 ๐‘‘โˆ’๐‘˜+2 5

and โˆฃ๐‘†โˆฃ โ‰ฅ

๐‘‘โˆ’๐‘˜+2 โˆฃ๐‘‰ (๐บ)โˆฃ. 2๐‘‘ โˆ’ ๐‘˜ + 2

Corollary 3.1. If ๐บ is a cubic graph then ๐œ“๐‘˜ (๐บ) โ‰ฅ ๐œ“3 (๐บ) โ‰ฅ 25 โˆฃ๐‘‰ (๐บ)โˆฃ.

5โˆ’๐‘˜ 8โˆ’๐‘˜ โˆฃ๐‘‰

(๐บ)โˆฃ, in particular

Corollary 3.2. Let ๐‘˜ โ‰ฅ 2 be a ๏ฌxed positive integer and ๐‘„๐‘‘ the ๐‘‘-dimensional hypercube. Then 1 ๐œ“๐‘˜ (๐‘„๐‘‘ ) = . lim ๐‘‘โ†’โˆž โˆฃ๐‘‰ (๐‘„๐‘‘ )โˆฃ 2 ๐‘‘โˆ’๐‘˜+2 Proof. By Theorem 3.2 we know that 2๐‘‘โˆ’๐‘˜+2 โˆฃ๐‘‰ (๐‘„๐‘‘ )โˆฃ โ‰ค ๐œ“๐‘˜ (๐‘„๐‘‘ ) โ‰ค ๐œ“2 (๐‘„๐‘‘ ) โ‰ค 1 2 โˆฃ๐‘‰ (๐‘„๐‘‘ )โˆฃ, and the result immediately follows.

Theorem 3.3. Let ๐‘˜ โ‰ฅ 2 and ๐‘‘ โ‰ฅ ๐‘˜ โˆ’ 1 be positive integers. There exists a ๐‘‘-regular graph ๐บ with ๐œ“๐‘˜ (๐บ) โ‰ค

๐‘‘โˆ’๐‘˜+2 โˆฃ๐‘‰ (๐บ)โˆฃ. 2๐‘‘ โˆ’ ๐‘˜ + 2

Proof. Given values of ๐‘˜ and ๐‘‘, a graph ๐บ may be constructed as follows. The vertex set is ๐‘‰ (๐บ) = ๐ด โˆช ๐ต with โˆฃ๐ดโˆฃ = (๐‘˜ โˆ’ 1)(๐‘‘ โˆ’ ๐‘˜ + 2) โˆฃ๐ตโˆฃ = ๐‘‘(๐‘˜ โˆ’ 1). Let the set ๐ด be an independent set in ๐บ and let ๐ต consist of ๐‘‘ components of ๐พ๐‘˜โˆ’1 in ๐บ. The remaining edges between ๐ด and ๐ต may be de๏ฌned arbitrarily to ๏ฌll ๐‘‘-regularity of ๐บ. Clearly, ๐ด is a ๐‘˜-path vertex cover of ๐บ since ๐ต does not contain a path on ๐‘˜ vertices and โˆฃ๐ดโˆฃ (๐‘˜ โˆ’ 1)(๐‘‘ โˆ’ ๐‘˜ + 2) ๐‘‘โˆ’๐‘˜+2 ๐œ“๐‘˜ (๐บ) โ‰ค = = . โˆฃ๐‘‰ (๐บ)โˆฃ โˆฃ๐ดโˆฃ + โˆฃ๐ตโˆฃ (๐‘˜ โˆ’ 1)(๐‘‘ โˆ’ ๐‘˜ + 2) + ๐‘‘(๐‘˜ โˆ’ 1) 2๐‘‘ โˆ’ ๐‘˜ + 2

4

Cartesian products

Recall that the Cartesian product ๐บโ–ก๐ป of graphs ๐บ = (๐‘‰ (๐บ), ๐ธ(๐บ)) and ๐ป = (๐‘‰ (๐ป), ๐ธ(๐ป)) has the vertex set ๐‘‰ (๐บ)ร—๐‘‰ (๐ป), and vertices (๐‘ข, ๐‘ฃ), (๐‘ฅ, ๐‘ฆ) are adjacent whenever ๐‘ข = ๐‘ฅ and ๐‘ฃ๐‘ฆ โˆˆ ๐ธ(๐ป), or ๐‘ข๐‘ฅ โˆˆ ๐ธ(๐บ) and ๐‘ฃ = ๐‘ฆ. By ๐‘๐บ and ๐‘๐ป we denote the natural projections to the factors ๐บ and ๐ป, respectively. For a ๏ฌxed vertex ๐‘ข โˆˆ ๐‘‰ (๐บ), the ๐ป-๏ฌber ๐‘ข ๐ป is the subgraph of ๐บโ–ก๐ป induced by the set of vertices {(๐‘ข, ๐‘ฃ), ๐‘ฃ โˆˆ ๐‘‰ (๐ป)}; analogously, for a ๏ฌxed vertex ๐‘ฃ โˆˆ ๐‘‰ (๐ป), the ๐บ-๏ฌber ๐บ๐‘ฃ is the subgraph of ๐บโ–ก๐ป induced by the set of vertices {(๐‘ข, ๐‘ฃ), ๐‘ข โˆˆ ๐‘‰ (๐บ)}. Note that each ๐ป-๏ฌber is isomorphic to ๐ป, and is an induced subgraph of ๐บโ–ก๐ป. Hence any vertex ๐‘˜-path cover ๐‘† of ๐บโ–ก๐ป must contains at least 6

๐œ“๐‘˜ (๐ป) vertices in every ๐ป-๏ฌber. Since there are โˆฃ๐‘‰ (๐บ)โˆฃ such ๏ฌbers, we get โˆฃ๐‘†โˆฃ โ‰ฅ โˆฃ๐‘‰ (๐บ)โˆฃ โ‹… ๐œ“๐‘˜ (๐ป). By reversing the roles of ๐บ and ๐ป we immediately get the following result. Proposition 4.1. Let ๐บ and ๐ป be graphs and ๐‘˜ โ‰ฅ 2 be a positive integer. Then ๐œ“๐‘˜ (๐บโ–ก๐ป) โ‰ฅ max{๐œ“๐‘˜ (๐บ) โ‹… โˆฃ๐‘‰ (๐ป)โˆฃ, โˆฃ๐‘‰ (๐บ)โˆฃ โ‹… ๐œ“๐‘˜ (๐ป)}. For ๐‘˜ = 2 this bound was observed already by Vizing [16], who expressed it in terms of independence numbers as follows ๐›ผ(๐บโ–ก๐ป) โ‰ค min{๐›ผ(๐บ) โˆฃ๐‘‰ (๐ป)โˆฃ , ๐›ผ(๐ป) โˆฃ๐‘‰ (๐บ)โˆฃ}.

(1)

A characterization of pairs of graphs that achieve the equality in (1) has been obtained in [9], and independently also in [5]. The characterization involves the existence of homomorphims of one factor to the so-called independence graph of the other factor. We will extend this result, by presenting the corresponding characterization of pairs of graphs that achieve the equality in 4.1. Recall that homomorphism from a graph ๐บ to a graph ๐ป is a mapping โ„Ž : ๐‘‰ (๐บ) โ†’ ๐‘‰ (๐ป) such that for any pair ๐‘ข, ๐‘ฃ โˆˆ ๐‘‰ (๐บ): if ๐‘ข๐‘ฃ โˆˆ ๐ธ(๐บ) then โ„Ž(๐‘ข)โ„Ž(๐‘ฃ) โˆˆ ๐ธ(๐ป). If there exists a homomorphism from a graph ๐บ to a graph ๐ป we write ๐บ โ†’ ๐ป. Let ๐บ be a graph, ๐‘˜ โ‰ฅ 2 an integer, and let ๐ด,๐ต be two subsets of ๐‘‰ (๐บ). We say that ๐ด, ๐ต form a good ๐‘˜-partition in ๐บ if there is no path on ๐‘˜ vertices in ๐บ of the form ๐‘Ž1 , . . . , ๐‘Ž๐‘™1 = ๐‘๐‘™1 , . . . , ๐‘๐‘™2 = ๐‘Ž๐‘™2 , . . . , ๐‘Ž๐‘˜ such that ๐‘Ž๐‘– โˆˆ ๐ด, ๐‘๐‘— โˆˆ ๐ต (for all applicable indices ๐‘– and ๐‘—); in addition, we allow that the path ends with ๐‘๐‘˜ and/or begins with ๐‘1 . In other words, there is no ๐‘˜ path in ๐ด โˆช ๐ต. In particular, any two ๐œ“๐‘˜ -sets ๐ด, ๐ต such that ๐ด โˆฉ ๐ต = โˆ… form a good ๐‘˜-partition of ๐บ. On the other hand, if one of the sets ๐ด, ๐ต is not a vertex ๐‘˜-path cover then ๐ด, ๐ต will clearly not form a good ๐‘˜-partition of ๐บ; indeed, if say ๐ด is not a vertex ๐‘˜-path cover then there is already a path of length ๐‘˜ in ๐‘‰ (๐บ) โˆ’ ๐ด that begins with ๐‘1 and ends with ๐‘๐‘˜ . We say that a vertex ๐‘˜-path cover of minimum size ๐œ“๐‘˜ (๐บ) is a ๐œ“๐‘˜ -set of ๐บ, and let ๐’ฎ be the set of all ๐œ“๐‘˜ -sets of ๐บ. Then the ๐œ“๐‘˜ -graph of ๐บ, denoted ฮจ๐‘˜ (๐บ) is the graph with ๐‘‰ (ฮจ๐‘˜ (๐บ)) = ๐’ฎ, and ๐ด, ๐ต โˆˆ ๐’ฎ are adjacent whenever they form a good ๐‘˜-partition in ๐บ. Theorem 4.1. Let ๐บ, ๐ป be arbitrary graphs. If ๐œ“๐‘˜ (๐บโ–ก๐ป) = max{๐œ“๐‘˜ (๐บ) โ‹… โˆฃ๐‘‰ (๐ป)โˆฃ, โˆฃ๐‘‰ (๐บ)โˆฃ โ‹… ๐œ“๐‘˜ (๐ป)} then ๐ป โ†’ ฮจ๐‘˜ (๐บ) or ๐บ โ†’ ฮจ๐‘˜ (๐ป). Proof. Suppose ๏ฌrst that ๐บ and ๐ป are graphs such that ๐œ“๐‘˜ (๐บ)โˆฃ๐‘‰ (๐ป)โˆฃ โ‰ฅ ๐œ“๐‘˜ (๐ป)โˆฃ๐‘‰ (๐บ)โˆฃ, and that the equality holds, thus ๐œ“๐‘˜ (๐บโ–ก๐ป) = ๐œ“๐‘˜ (๐บ)โˆฃ๐‘‰ (๐ป)โˆฃ. 7

Let ๐ด be a ๐œ“๐‘˜ -set of ๐บโ–ก๐ป. Note that in each ๐บ-๏ฌber of ๐บโ–ก๐ป at least ๐œ“๐‘˜ (๐บ) vertices are from ๐ด. Combining this with โˆฃ๐ดโˆฃ = ๐œ“๐‘˜ (๐บ)โˆฃ๐‘‰ (๐ป)โˆฃ we ๏ฌnd that in each ๐บ-๏ฌber there are exactly ๐œ“๐‘˜ (๐บ) vertices from ๐ด. In other words, vertices of ๐ด in each ๐บ-๏ฌber form a ๐œ“๐‘˜ -set of this ๐บ-๏ฌber. Let ๐‘ข, ๐‘ฃ โˆˆ ๐‘‰ (๐ป) be adjacent vertices. Then we claim that the sets ๐‘๐บ (๐ดโˆฉ๐บ๐‘ข ) and ๐‘๐บ (๐ดโˆฉ๐บ๐‘ฃ ) are two ๐œ“๐‘˜ -sets of that form a good ๐‘˜-partition of ๐บ. Indeed, otherwise there would exist a path of length ๐‘˜ in ๐บ of the form ๐‘Ž1 , . . . , ๐‘Ž๐‘™1 = ๐‘๐‘™1 , . . . , ๐‘๐‘™2 = ๐‘Ž๐‘™2 , . . . , ๐‘Ž๐‘˜ as described above, and hence in ๐บโ–ก๐ป โˆ’๐ด there would be a path (๐‘Ž1 , ๐‘ข), . . . , (๐‘Ž๐‘™1 , ๐‘ข), (๐‘๐‘™1 , ๐‘ฃ), . . . , (๐‘๐‘™2 , ๐‘ฃ), (๐‘Ž๐‘™2 , ๐‘ข), . . . , (๐‘Ž๐‘˜ , ๐‘ข) of the same length ๐‘˜. This is a contradiction with ๐ด being a vertex ๐‘˜path cover of ๐บ. Hence the mapping โ„Ž : ๐ป โ†’ ฮจ๐‘˜ (๐บ), de๏ฌned by โ„Ž(๐‘ข) = ๐‘๐บ (๐ด โˆฉ ๐บ๐‘ข ) is a homomorphism, since for any ๐‘ข๐‘ฃ โˆˆ ๐ธ(๐ป), also โ„Ž(๐‘ข) and โ„Ž(๐‘ฃ) are adjacent. Note that for ๐‘˜ = 2 the converse of the statement of Theorem 4.1 is also true. We get nicer results if we simplify the structure of one of the factors. If say, ๐ป is isomorphic to the complete graph ๐พ๐‘› , then ฮจ๐‘˜ (๐ป) has a nice structure; its vertices consist of all ๐‘˜-combinations of the set {1, . . . , ๐‘›}, and two such ๐‘˜-combinations are adjacent whenever their union is {1, . . . , ๐‘›}. Clearly ฮจ๐‘˜ (๐พ๐‘› ) is nontrivial when 2๐‘˜ โ‰ฅ ๐‘›. The part from Theorem 4.1, i.e. ๐ป โ†’ ฮจ๐‘˜ (๐พ๐‘› ), thus translates to the colorings of a graph ๐ป with ๐‘˜combinations. The following theorem, containing the formulas for ๐œ“๐‘˜ of grids (Cartesian products of paths), is the main result of this section. Theorem 4.2.

(i) ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ ) = 2๐‘›๐‘˜ + โŒŠ 2๐‘˜ 3 โŒ‹, where ๐‘›, ๐‘˜ โ‰ฅ 1,

(ii) ๐œ“3 (๐‘ƒ2๐‘› โ–ก ๐‘ƒ2๐‘˜ ) = 2๐‘›๐‘˜, where ๐‘›, ๐‘˜ โ‰ฅ 1, (iii) ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 ) = ๐‘›(2๐‘˜ + 1) + โŒŠ 2๐‘˜+1 3 โŒ‹, where 1 โ‰ค ๐‘› โ‰ค ๐‘˜. Proof. (i) Let us label the vertices of ๐‘ƒ2๐‘›+1 consecutively by ๐‘ข1 , ๐‘ข2 , . . . , ๐‘ข2๐‘›+1 and the vertices of ๐‘ƒ2๐‘˜ by ๐‘ฃ1 , ๐‘ฃ2 , . . . , ๐‘ฃ2๐‘˜ . With this labeling, the vertex ๐‘ฃ๐‘— (๐‘ข๐‘– , ๐‘ฃ๐‘— ) โˆˆ ๐‘‰ (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ ) is in ๐‘–th ๏ฌber ๐‘ข๐‘– ๐‘ƒ2๐‘˜ and in ๐‘— th ๏ฌber ๐‘ƒ2๐‘›+1 . 2๐‘˜ Let us ๏ฌrst prove that ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ ) โ‰ค 2๐‘›๐‘˜ + โŒŠ 3 โŒ‹. We construct a vertex 3-path cover set ๐‘† in the following way: ๐‘† = {(๐‘ข2๐‘–+1 , ๐‘ฃ3๐‘—+3 ), (๐‘ข2๐‘– , ๐‘ฃ3๐‘—+1 ), (๐‘ข2๐‘– , ๐‘ฃ3๐‘—+2 )โˆฃ for applicable indices ๐‘– and ๐‘—} . It is not di๏ฌƒcult to verify that ๐‘† is a vertex 3-path cover set (Fig. 1) and gives the desired upper bond.

8

v1 v2 v3 v4 v5 v6 v 3k 2 v 3k 1 v 3k u1 u2 u3 u4 u5

u2n 1 u2n u2n 1

Figure 1: A vertex 3-path covering of ๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ Now let us prove that ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ ) โ‰ฅ 2๐‘›๐‘˜ + โŒŠ 2๐‘˜ 3 โŒ‹. Let ๐‘† be the ๐‘ข ๐‘– optimal vertex 3-path covering. Consider the ๏ฌber ๐‘ƒ2๐‘˜ , for some ๐‘– โˆˆ {1, 2, . . . , 2๐‘›}. Suppose (๐‘ข๐‘– , ๐‘ฃ๐‘— ) โˆ•โˆˆ ๐‘†, for some ๐‘— โˆˆ {1, 2, . . . , 2๐‘˜ โˆ’1}, and all its neighbors ๐‘ข in ๐‘– ๐‘ƒ2๐‘˜ are in ๐‘†. Then in the ๏ฌber ๐‘ข๐‘–+1 ๐‘ƒ2๐‘˜ either the vertex (๐‘ข๐‘–+1 , ๐‘ฃ๐‘— ) is in ๐‘† (left-hand side of Fig. 2) or all its neighbors in this ๏ฌber are in ๐‘† (right-hand side of Fig. 2).

vj

vj

ui ui 1

ui ui 1

Figure 2: Vertex (๐‘ข๐‘– , ๐‘ฃ๐‘— ) is not in ๐‘† Suppose (๐‘ข๐‘– , ๐‘ฃ๐‘— ) and (๐‘ข๐‘– , ๐‘ฃ๐‘—+1 ), for some ๐‘— โˆˆ {1, 2, . . . , 2๐‘˜ โˆ’ 1}, are not in ๐‘†. Then the vertices (๐‘ข๐‘–+1 , ๐‘ฃ๐‘— ) and (๐‘ข๐‘–+1 , ๐‘ฃ๐‘—+1 ) must be in ๐‘†, otherwise ๐‘† is not a vertex 3-path covering (see Fig. 3). Let ๐ด be the set of all vertices in ๏ฌber ๐‘ข๐‘– ๐‘ƒ2๐‘˜ that are not in ๐‘†, and ๐ต the set of all vertices in ๏ฌber ๐‘ข๐‘–+1 ๐‘ƒ2๐‘˜ that are in ๐‘†. We shall prove that โˆฃ๐ดโˆฃ โ‰ค โˆฃ๐ตโˆฃ by ๏ฌnding a one to one function ๐‘“ from ๐ด to ๐ต. Let (๐‘ข๐‘– , ๐‘ฃ๐‘— ) be a vertex in ๐ด. Suppose that the vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜ ) is not in ๐ด. By Fig. 2 and Fig. 3 we set ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ๐‘— )) = (๐‘ข๐‘–+1 , ๐‘ฃ๐‘— ) if (๐‘ข๐‘–+1 , ๐‘ฃ๐‘— ) is in ๐‘†, otherwise we set ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ๐‘— )) = (๐‘ข๐‘–+1 , ๐‘ฃ๐‘—+1 ). This is obviously a one to one function. We take a di๏ฌ€erent approach if the vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜ ) is in ๐ด. If also the vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’1 ) is in ๐ด we can use the same argument then above since 9

vj vj 1

ui ui 1 Figure 3: Vertices (๐‘ข๐‘– , ๐‘ฃ๐‘— ) and (๐‘ข๐‘– , ๐‘ฃ๐‘—+1 ) are not in ๐‘† according to Fig. 3 we can set ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ2๐‘˜ )) = (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜ ). If the vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’1 ) โˆˆ / ๐ด and (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜ ) โˆˆ ๐ต then we can set ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ2๐‘˜ )) = (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜ ) otherwise we can set ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ2๐‘˜ )) = (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’1 ). It can happen that the vertex (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’1 ) is already an image of the vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’2 ). If this situation occurs we can set ether ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’2 )) = (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’2 ) or ๐‘“ ((๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’2 )) = (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’3 ) since at least one of the vertices (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’2 ) or (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’3 ) is in ๐ต. Note that if vertex (๐‘ข๐‘– , ๐‘ฃ2๐‘˜โˆ’2 ) exists in the grid then so must vertex (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’3 ) since every ๐‘ƒ2๐‘˜ -๏ฌber has even number of vertices. It can occur that the vertex (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘˜โˆ’3 ) is also an image of a vertex in ๏ฌber ๐‘ข๐‘– ๐‘ƒ2๐‘˜ . In this case we can repeat the above procedure. We can always ๏ฌnd a one to one function from ๐ด to ๐ต and hence โˆฃ๐ดโˆฃ โ‰ค โˆฃ๐ตโˆฃ. Therefore two consecutive ๏ฌbers, ๐‘ข๐‘– ๐‘ƒ2๐‘˜ and ๐‘ข๐‘–+1 ๐‘ƒ2๐‘˜ , must have at least 2๐‘˜ vertices in ๐‘†. There are ๐‘› such paired ๏ฌbers in ๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜ . Together with the additional ๏ฌber, which is isomorphic to the path ๐‘ƒ2๐‘˜ , there must be at lest 2๐‘›๐‘˜ + โŒŠ 2๐‘˜ 3 โŒ‹ vertices in ๐‘†. (ii) We follow the same line of thought as in (i), with the exception that in this case there is no additional ๏ฌber.

(iii) Note that in the proof of (i), two consecutive ๏ฌbers, ๐‘ข๐‘– ๐‘ƒ2๐‘˜ and ๐‘ข๐‘–+1 ๐‘ƒ , have at least 2๐‘˜ vertices in ๐‘†. The ๏ฌber ๐‘ข๐‘– ๐‘ƒ 2๐‘˜ 2๐‘˜+1 in this case has odd number of vertices and it can occur that two consecutive ๏ฌbers do not have at least 2๐‘˜ + 1 vertices in ๐‘†. Nevertheless, two consecutive ๏ฌbers, ๐‘ข๐‘– ๐‘ƒ ๐‘ข๐‘–+1 ๐‘ƒ 2๐‘˜+1 and 2๐‘˜+1 , must still have at least 2๐‘˜ vertices in ๐‘†. Moreover if they have exactly 2๐‘˜ vertices in ๐‘†, then for ๏ฌbers ๐‘ข๐‘– ๐‘ƒ2๐‘˜+1 and ๐‘ข๐‘–+1 ๐‘ƒ2๐‘˜+1 the vertices (๐‘ข๐‘– , ๐‘ฃ2๐‘— ) and (๐‘ข๐‘–+1 , ๐‘ฃ2๐‘— ), ๐‘— โˆˆ {1, . . . , ๐‘˜}, must all be in ๐‘† (see the structure of sets ๐ด and ๐ต in (i)). Hence, both ๏ฌbers ๐‘ข๐‘– ๐‘ƒ2๐‘˜+1 and ๐‘ข๐‘–+1 ๐‘ƒ2๐‘˜+1 each contribute at least ๐‘˜ vetices to ๐‘†. Having done the basic observation let us proceed with the proof. For the upper bound the same construction as in (i) su๏ฌƒces. Now we want to prove that ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 ) โ‰ฅ ๐‘›(2๐‘˜ + 1) + โŒŠ 2๐‘˜+1 3 โŒ‹. Asumme that 0 โ‰ค ๐‘› โ‰ค ๐‘˜ and ๐‘† is the optimal vertex 3-path covering of the graph ๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 . We proceed with induction on ๐‘›. For ๐‘› = 0, the graph ๐‘ƒ1 โ–ก ๐‘ƒ2๐‘˜+1 is isomorphic to the path ๐‘ƒ2๐‘˜+1 , and hence ๐œ“3 (๐‘ƒ2๐‘˜+1 ) = โŒŠ 2๐‘˜+1 3 โŒ‹. In the induction step we 10

consider two cases. Suppose that the last two ๏ฌbers, ๐‘ข2๐‘› ๐‘ƒ2๐‘˜+1 and ๐‘ข2๐‘›+1 ๐‘ƒ2๐‘˜+1 , contribute at least 2๐‘˜ + 1 vertices to ๐‘†. Assume that ๐œ“3 (๐‘ƒ2๐‘›โˆ’1 โ–ก ๐‘ƒ2๐‘˜+1 ) = (๐‘› โˆ’ 1)(2๐‘˜ + 2๐‘˜+1 1) + โŒŠ 2๐‘˜+1 3 โŒ‹ and ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 ) < ๐‘›(2๐‘˜ + 1) + โŒŠ 3 โŒ‹. If we remove the ๐‘ข ๐‘ข ๏ฌbers, 2๐‘› ๐‘ƒ2๐‘˜+1 and 2๐‘›+1 ๐‘ƒ2๐‘˜+1 , we remove at least 2๐‘˜ + 1 vertices from ๐‘† an therefore ๐œ“3 (๐‘ƒ2๐‘›โˆ’1 โ–ก ๐‘ƒ2๐‘˜+1 ) < ๐‘›(2๐‘˜ + 1) + โŒŠ 2๐‘˜+1 3 โŒ‹ โˆ’ 2๐‘˜ โˆ’ 1. By induction assumption it follows that โŒŠ โŒ‹ โŒŠ โŒ‹ 2๐‘˜ + 1 2๐‘˜ + 1 (๐‘› โˆ’ 1)(2๐‘˜ + 1) + < ๐‘›(2๐‘˜ + 1) + โˆ’ 2๐‘˜ โˆ’ 1 . 3 3 By doing some basic calculation this leads us to a contradiction. For the second case assume that the last two ๏ฌbers, ๐‘ข2๐‘› ๐‘ƒ2๐‘˜+1 and ๐‘ข2๐‘›+1 ๐‘ƒ2๐‘˜+1 , contribute exactly 2๐‘˜ vertices to ๐‘†. According to the observation above, if we remove the last ๏ฌber ๐‘ข2๐‘›+1 ๐‘ƒ2๐‘˜+1 from the graph ๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 we remove at least ๐‘˜ vertices from ๐‘†. Suppose again that ๐œ“3 (๐‘ƒ2๐‘›+1 โ–ก ๐‘ƒ2๐‘˜+1 ) < 2๐‘˜+1 ๐‘›(2๐‘˜ + 1) + โŒŠ 2๐‘˜+1 3 โŒ‹. Then ๐œ“3 (๐‘ƒ2๐‘› โ–ก ๐‘ƒ2๐‘˜+1 ) < ๐‘›(2๐‘˜ + 1) + โŒŠ 3 โŒ‹ โˆ’ ๐‘˜. But 2๐‘› according to (i) we know that ๐œ“3 (๐‘ƒ2๐‘› โ–ก ๐‘ƒ2๐‘˜+1 ) = 2๐‘›๐‘˜ + โŒŠ 3 โŒ‹ and hence โŒŠ โŒ‹ โŒŠ โŒ‹ 2๐‘› 2๐‘˜ + 1 2๐‘›๐‘˜ + < ๐‘›(2๐‘˜ + 1) + โˆ’๐‘˜. 3 3 By doing some basic calculation we get that ๐‘˜ < ๐‘›, which again is a contradiction since ๐‘› โ‰ค ๐‘˜. As seen in Theorem 4.2, it is already hard to determine the exact value of ๐œ“3 for grids therefore it would be nice to have at least some lower or upper bounds for ๐œ“๐‘˜ , ๐‘˜ โ‰ฅ 4. Lemma 4.1. For each ๐‘™ โ‰ฅ 2, ๐œ“๐‘™2 (๐‘ƒ2๐‘™ โ–ก ๐‘ƒ3๐‘™ ) โ‰ฅ ๐‘™. Proof. Assume to the contrary that ๐‘† is a ๐‘˜-path vertex cover of ๐‘ƒ2๐‘™ โ–ก ๐‘ƒ3๐‘™ , with โˆฃ๐‘†โˆฃ โ‰ค ๐‘™ โˆ’ 1. Then ๐‘ƒ2๐‘™ โ–ก ๐‘ƒ3๐‘™ has at least ๐‘™ ๐‘ƒ3๐‘™ -layers not containing any vertex of ๐‘†. Denote such layers by ๐‘ƒ๐‘“1 , ๐‘ƒ๐‘“2 , . . . , ๐‘ƒ๐‘“๐‘™ . Since โˆฃ๐‘†โˆฃ โ‰ค ๐‘™ โˆ’ 1, then two consecutive layers can be connected vertically in ๏ฌrst and last thirds without crossing ๐‘†. Formally, for each ๐‘– โˆˆ {1, . . . , (๐‘™ โˆ’ 1)} there exists ๐‘–๐‘Ž โˆˆ {1, . . . , ๐‘™} and ๐‘–๐‘ โˆˆ {2๐‘™ + 1, . . . , 3๐‘™} such that for each โ„Ž โˆˆ {๐‘Ž๐‘– , . . . , ๐‘Ž๐‘–+1 } the vertices (๐‘–๐‘Ž , โ„Ž), (๐‘–๐‘ , โ„Ž) โˆˆ / ๐‘†. Finally, we construct a path ๐‘ƒ moving horizontally on ๐‘ƒ๐‘“1 , ๐‘ƒ๐‘“2 , . . . , ๐‘ƒ๐‘“๐‘™ layers and vertically on ๐‘–๐‘Ž , ๐‘–๐‘ columns: ๐‘ƒ =(1, ๐‘“1 ), . . . , (1๐‘ , ๐‘“1 ), (1๐‘ , ๐‘“2 ), . . . , (2๐‘Ž , ๐‘“2 ), (2๐‘Ž , ๐‘“3 ), . . . , (3๐‘ , ๐‘“3 ), (3๐‘ , ๐‘“4 ), . . . , (4๐‘Ž , ๐‘“4 ), . . . . Such a path contains at least ๐‘™ vertices in each layer ๐‘ƒ๐‘“1 , ๐‘ƒ๐‘“2 , . . . , ๐‘ƒ๐‘“๐‘™ , hence the total order of the path ๐‘ƒ is at least ๐‘™2 , a contradiction. 11

First we present a lower bound with the help of Lemma 4.1. โˆš โˆš Proposition 4.2. For ๐‘˜ โ‰ฅ 4, ๐‘› โ‰ฅ 2 ๐‘˜, ๐‘š โ‰ฅ 3 ๐‘˜, the following holds ๐‘›๐‘š โˆš โ‰ค ๐œ“๐‘˜ (๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š ). 24 ๐‘˜ Proof. We split the whole graph ๐‘ƒ๐‘› โˆš โ–ก ๐‘ƒ๐‘š โˆš into ๐‘Ÿ disjoint subgraphs isomorphic to ๐‘ƒ2โˆš๐‘˜ โ–ก ๐‘ƒ3โˆš๐‘˜ such that ๐‘Ÿ(2 ๐‘˜)(3 ๐‘˜) โ‰ฅ 41 ๐‘›๐‘š. By Lemma 4.1 a โˆš ๐‘˜-path vertex cover must have at least ๐‘˜ vertices in each subgraph isomorphic to ๐‘ƒ2โˆš๐‘˜ โ–ก ๐‘ƒ3โˆš๐‘˜ in ๐บ, hence: โˆš ๐‘›๐‘š ๐œ“๐‘˜ (๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š ) โ‰ฅ ๐‘Ÿ ๐‘˜ โ‰ฅ โˆš . 24 ๐‘˜

We conclude the paper with the following upper bound. Proposition 4.3. For ๐‘˜ โ‰ฅ 4 the following holds 2๐‘›๐‘š ๐‘›๐‘š . ๐œ“๐‘˜ (๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š ) โ‰ค โˆš โˆ’ ๐‘˜ ๐‘˜ โˆš Proof. We will construct a ๐‘˜-path vertex cover with at most 2๐‘›๐‘š โˆ’ ๐‘›๐‘š ๐‘˜ ๐‘˜ { } โˆš vertices. Let ๐‘†1 = (๐‘–, ๐‘—) โˆˆ ๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š โˆฃ ๐‘– โ‰ก 0 (modโŒŠ ๐‘˜โŒ‹) (for all applicable { โˆš } indicies ๐‘– and ๐‘—) and similarly ๐‘†2 = (๐‘–, ๐‘—) โˆˆ ๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š โˆฃ ๐‘— โ‰ก 0 (modโŒŠ ๐‘˜โŒ‹) . It is easy to see that ๐‘† = (๐‘†1 โˆช ๐‘†2 )โˆ–(๐‘†1 โˆฉ ๐‘†2 ) is a ๐‘˜-path vertex cover, since the largest subgraph of ๐‘ƒ๐‘› โ–ก ๐‘ƒ๐‘š with all vertices uncovered is isomorphic to ๐‘ƒโŒŠโˆš๐‘˜โŒ‹โˆ’1 โ–ก ๐‘ƒโŒŠโˆš๐‘˜โŒ‹โˆ’1 . โˆš In a ๐‘ƒ๐‘› -layer we cover each โŒŠ ๐‘˜โŒ‹-th vertex, since there ๐‘š layers the โˆš . Similarly, โˆฃ๐‘†2 โˆฃ โ‰ค ๐‘›๐‘š โˆš . The vertices size of ๐‘†1 is at most โˆฃ๐‘†1 โˆฃ โ‰ค โŒŠ๐‘›๐‘š ๐‘˜โŒ‹ โŒŠ ๐‘˜โŒ‹

(๐‘–, ๐‘—) โˆˆ ๐‘†1 โˆฉ ๐‘†2 can be left uncovered, since all the vertices (๐‘– ยฑ 1, ๐‘—) and (๐‘–, ๐‘— ยฑ 1) are in ๐‘†. Since the size of โˆฃ๐‘†1 โˆฉ ๐‘†2 โˆฃ โ‰ค ๐‘›๐‘š ๐‘˜ , the assertion follows.

References [1] V. E. Alekseev, R. Boliac, D. V. Korobitsyn, V. V. Lozin, NP-hard graph problems and boundary classes of graphs, Theor. Comp. Science 389 (1-2) (2007) 219โ€“236. [2] R. Boliac, K. Cameron, V.V. Lozin, On computing the dissociation number and the induced matching number of bipartite graphs, Ars Combin. 72 (2004) 241โ€“253.

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[3] M. Borowiecki, I. Broere, M. Frick, P. Mihยดok and G. Semaniห‡sin, Survey of hereditary properties of graphs, Discuss. Math. - Graph Theory 17 (1997) 5โ€“50. [4] B. Breห‡sar, F. Kardoห‡s, J. Katreniห‡c, G. Semaniห‡sin, Minimum ๐‘˜-path vertex cover, Discrete Appl. Math. 159 (12) (2011) 1189โ€“1195. 1189โ€“1195. [5] B. Breห‡sar and B. Zmazek, On the independence graph of a graph, Discrete Math. 272 (2001), 263โ€“268. [6] K. Cameron, P. Hell, Independent packings in structured graphs, Math. Program. 105 (2-3) (2006) 201โ€“213. [7] P. Erdหos and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959) 337โ€“356. [8] F. Gยจoring, J. Harant, D. Rautenbach and I. Schiermeyer, On Findependence in graphs, Discuss. Math. - Graph Theory 29 (2) (2009) 377โ€“383. [9] P. Hell, X. Yu, H. Zhou, Independence ratios of graph powers, Discrete Math. 127 (1994) 213โ€“220. [10] F. Kardoห‡s, J. Katreniห‡c, I. Schiermeyer, On computing the minimum 3path vertex cover and dissociation number of graphs, to apper in Theor. Comp. Science. [11] L. Lovยดasz, On decompositions of graphs, Studia Sci. Math Hungar. 1 (1966) 237โ€“238. [12] V. V. Lozin, D. Rautenbach, Some results on graphs without long induced paths, Inf. Process. Lett. 88 (4) (2003) 167โ€“171. [13] M. Novotnยด y, Design and Analysis of a Generalized Canvas Protocol. (to appear in Proceedings of WISTP 2010: Information Security Theory and Practice. Security and Privacy of Pervasive Systems and Smart Devices, LNCS 6033, Springer 2010). [14] Y. Orlovich, A. Dolguib, G. Finkec, V. Gordond, F. Wernere, The complexity of dissociation set problems in graphs, Discrete Appl. Math. 159 (13) (2011) 1352โ€“1366. [15] J. Tua, W. Zhoub, A factor 2 approximation algorithm for the vertex cover P3 problem, Information Processing Letters 111 (14) (2011) 683โ€“ 686 [16] V.G. Vizing, The Cartesian product of graphs, Vychisl. Sistemy 9 (1963), 30โ€“43. 13

[17] M. Yannakakis, Node-deletion problems on bipartite graphs, SIAM J. Computing 10 (1981) 310โ€“327.

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