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 Boหstjan Breหsarโ, Marko Jakovacโ Faculty of Natural Science and Mathematics, University of Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Jยดan Katreniหcโ and Gabriel Semaniหsinโก
ห arik University, Koหsice, 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 ๐บ is called a ๐-path vertex cover if every path of order ๐ in ๐บ contains at least one vertex from ๐. Denote by ๐๐ (๐บ) the minimum cardinality of a ๐-path vertex cover in ๐บ. In this paper present an upper bound for ๐3 of graphs with given average degree. We also give a lower bound for ๐๐ of regular graphs. For the Cartesian products of two paths we give an asymptotically tight bound for ๐๐ and the exact value for ๐3 .
1
Introduction
Let ๐บ be a graph and ๐ be a positive integer. Then ๐ โ ๐ (๐บ) is the vertex ๐-path cover of ๐บ if every path on ๐ vertices in ๐บ contains a vertex from ๐. We denote by ๐๐ (๐บ) the cardinality of a minimum vertex ๐-path cover in ๐บ. 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]. โ
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. โก 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, ๐2 (๐บ) equals to the size of minimum vertex cover, and ๐2 (๐บ) = โฃ๐ (๐บ)โฃ โ ๐ผ(๐บ) where ๐ผ(๐บ) stands for the independence number. The value of ๐3 (๐บ) corresponds to the concept of dissociation number of a graph [17], de๏ฌned as follows. A subset of vertices in a graph ๐บ is called dissociation if it induces a subgraph with maximum degree 1. The number of vertices in a maximum cardinality dissociation set in ๐บ is called the dissociation number of ๐บ and is denoted ๐๐๐ ๐ (๐บ). Clearly ๐3 (๐บ) = โฃ๐ (๐บ)โฃ โ ๐๐๐ ๐ (๐บ). The problem of computing ๐๐๐ ๐ (๐บ) 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 ๐3 (๐บ) in running time ๐(1.5171๐ ) for a graph of order ๐ was presented in [10]. The problem of computing ๐๐ (๐บ) is NP-hard for each ๐ โฅ 2, but polynomial for instance in trees, as shown in [4]. The authors investigate upper bounds on the value of ๐๐ (๐บ) and provide several estimations and exact values of ๐๐ (๐บ). It is also proven that ๐3 (๐บ) โค (2๐ + ๐)/6, for every graph ๐บ with ๐ vertices and ๐ edges. In this paper we present a more general result showing that for an arbitrary graph ๐บ with ๐ vertices and ๐ edges and an integer ๐ such that ๐๐ ๐ ๐โค๐ ๐ โค ๐ + 1 we have ๐3 (๐บ) โค ๐+2 + (๐+1)(๐+2) . In Section 3 we consider bounds on ๐-regular graphs and show that for an arbitrary integer ๐ โฅ 2 and ๐โ๐+2 ๐-regular graph ๐บ, ๐ โฅ ๐ โ 1 we have ๐๐ (๐บ) โฅ 2๐โ๐+2 โฃ๐ (๐บ)โฃ. 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 ๐๐ . Proposition 1.1. Let ๐ โฅ 2 and ๐ โฅ ๐ be positive integers. Then โ โ โ ๐๐ (๐๐ ) = ๐๐ , โ ๐๐ (๐ถ๐ ) = โ ๐๐ โ, โ ๐๐ (๐พ๐ ) = ๐ โ ๐ + 1.
2
Upper bounds on degree of vertices
First, recall the result of [4] which is a consequence of Lovยดaszโs decomposition [11] of a graph with maximum degree ฮ into subgraphs of maximum degree 1. 2
Lemma 2.1 ([4]). Let ๐บ be a graph of maximum degree ฮ. Then ๐3 (๐บ) โค
โ ฮโ1 2 โ โ ฮ+1 2 โ
โฃ๐ (๐บ)โฃ.
In the following theorem we give a tight upper bound in terms of average degree of graph, which extends the result ๐3 (๐บ) โค (2๐ + ๐)/6 of [4]. Theorem 2.1. Let ๐บ be a graph of order ๐, size ๐ and average degree ๐(๐บ), and ๐ be a positive integer such that 2๐ โค ๐(๐บ) โค 2๐ + 2. Then ๐3 (๐บ) โค
๐๐ ๐ + 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 โ 1)๐ ๐ ๐3 (๐บ) โค + ๐+1 ๐(๐ + 1) for all graphs with average degree ๐, 2๐ โ 2 โค ๐ โค 2๐. We aim to prove that for a graph ๐บ with average degree ๐, 2๐ โค ๐ โค 2๐ + 2 the following holds ๐๐ ๐ ๐3 (๐บ) โค + . ๐ + 2 (๐ + 1)(๐ + 2) Systematically remove from ๐บ vertices of degree at least 2(๐ + 1). We get a graph ๐บโฒ of ฮ(๐บโฒ ) โค 2(๐ + 1) โ 1. For ๐บโฒ we now look at two cases. In the ๏ฌrst case we do not use the induction step. If ๐๐โฒ โค ๐โฒ , then by Lemma 2.1 ๐3 (๐บโฒ ) โค = = โค
๐ ๐โฒ ๐+1 ๐(๐ + 2) ๐โฒ (๐ + 1)(๐ + 2) ๐(๐ + 1) + ๐ โฒ ๐ (๐ + 1)(๐ + 2) ๐ ๐โฒ ๐โฒ + ๐+2 (๐ + 1)(๐ + 2)
In the second case, if ๐๐โฒ โฅ ๐โฒ we use the induction step. ๐3 (๐บโฒ ) โค = โค =
๐โ1 โฒ ๐ + ๐+1 ๐โ1 โฒ ๐ + ๐+1 ๐โ1 โฒ ๐ + ๐+1 ๐ ๐โฒ + ๐+2
๐โฒ ๐(๐ + 1) 2๐โฒ ๐โฒ + ๐(๐ + 1)(๐ + 2) (๐ + 1)(๐ + 2) 2๐โฒ ๐โฒ + (๐ + 1)(๐ + 2) (๐ + 1)(๐ + 2) ๐โฒ (๐ + 1)(๐ + 2) 3
Next we prove a lower bound for ๐3 (๐บ), showing that the upper bound of Theorem 2.1 is tight. Theorem 2.2. For every positive rational number ๐๐ > 1, 2๐ โค there exists a graph ๐บ with average degree ๐(๐บ) = ๐๐ and ๐3 (๐บ) โฅ
๐ ๐
โค 2๐ + 2,
๐๐ ๐ + . ๐ + 2 (๐ + 1)(๐ + 2)
Proof. Denote by ๐ป๐ a complete graph on ๐ vertices minus one perfect matching (assume ๐ is even). Clearly โฃ๐ (๐ป๐ )โฃ = ๐ and โฃ๐ธ(๐ป๐ )โฃ =
๐(๐ โ 1) ๐ ๐(๐ โ 2) โ = . 2 2 2
Clearly ๐๐๐ ๐ (๐ป๐ ) โค 2, because arbitrary three vertices of ๐ป๐ forms a path of order three. Therefore ๐3 (๐ป๐ ) = ๐ โ 2, for ๐ โฅ 2. We construct the graph ๐บ as the disjoint union of โ ๐ฅ components ๐ป2๐+2 , โ ๐ฆ components ๐ป2๐+4 . We let ๐ฅ = (2๐ โ ๐ + 2๐๐)(๐ + 2) and ๐ฆ = (๐ โ 2๐๐)(๐ + 1). First, we verify the average degree of the graph ๐บ. There are ๐ฅ(2๐ + 2) vertices of degree 2๐ and ๐ฆ(2๐ + 4) vertices of degree 2๐ + 2, hence we get ๐(๐บ) = = = = =
๐ฅ(2๐ + 2)(2๐) + ๐ฆ(2๐ + 4)(2๐ + 2) ๐ฅ(2๐ + 2) + ๐ฆ(2๐ + 4) (2๐ โ ๐ + 2๐๐)(๐ + 2)(2๐ + 2)(2๐) + (๐ โ 2๐๐)(๐ + 1)(2๐ + 4)(2๐ + 2) (2๐ โ ๐ + 2๐๐)(๐ + 2)(2๐ + 2) + (๐ โ 2๐๐)(๐ + 1)(2๐ + 4) (2๐ โ ๐ + 2๐๐)(2๐) + (๐ โ 2๐๐)(2๐ + 2) (2๐ โ ๐ + 2๐๐) + (๐ โ 2๐๐) 2๐๐ โ (๐ + 2๐๐)(2๐) + (๐ โ 2๐๐)(2๐) + 2(๐ โ 2๐๐) 2๐ ๐ . ๐
Now, let us verify the value of ๐3 of ๐บ. Clearly, ๐3 (๐บ) = ๐ฅ โ
๐3 (๐ป2๐+2 ) + ๐ฆ โ
๐3 (๐ป2๐+4 ) = ๐ฅ(2๐) + ๐ฆ(2๐ + 2).
4
And this exactly matches the value of the formula ๐3 (๐บ) = = = = =
3
๐๐ ๐ + ๐ + 2 (๐ + 1)(๐ + 2) ๐(๐ฅ(2๐ + 2) + ๐ฆ(2๐ + 4)) ๐ฅ(2๐)(2๐ + 2) + ๐ฆ(2๐ + 2)(2๐ + 4) + ๐+2 2(๐ + 1)(๐ + 2) ๐๐ฅ(2๐ + 2) ๐ฅ(2๐) + ๐ฆ(2๐ + 4) + 2๐๐ฆ + ๐+2 (๐ + 2) ๐๐ฅ(2๐ + 2) ๐ฅ(2๐) + 2๐๐ฆ + + 2๐ฆ ๐+2 (๐ + 2) 2๐๐ฅ + 2๐๐ฆ + 2๐ฆ.
Regular graphs
In the main theorem of this section we shall use the following result of Erdหos and Gallai [7]. Theorem 3.1 ([7]). If ๐บ is a graph on ๐ vertices that does not contain a path of order ๐ then it cannot have more than ๐(๐โ2) edges. Moreover, the 2 bound is achieved when the graph consists of disjoint complete graphs on ๐ โ 1 vertices. Theorem 3.2. Let ๐ โฅ 2 and ๐ โฅ ๐ โ 1 be positive integers. Then for any ๐-regular graph ๐บ the following holds: ๐๐ (๐บ) โฅ
๐โ๐+2 โฃ๐ (๐บ)โฃ. 2๐ โ ๐ + 2
Proof. Let ๐ โ ๐ (๐บ) be a vertex ๐-path cover and ๐ = ๐ (๐บ) โ ๐. Let ๐ธ๐ , ๐ธ๐ be the number of edges with both endvertices in ๐ and ๐ respectively. Let ๐ธ๐๐ be the number of edges with one endvertex in ๐ and the second vertex in ๐ . Then obviously โฃ๐ธ(๐บ)โฃ = 21 ๐โฃ๐ (๐บ)โฃ = โฃ๐ธ๐ โฃ + โฃ๐ธ๐๐ โฃ + โฃ๐ธ๐ โฃ. Since ๐บ is ๐-regular, ๐โฃ๐โฃ = 2โฃ๐ธ๐ โฃ + โฃ๐ธ๐๐ โฃ. Therefore โฃ๐โฃ โฅ ๐1 โฃ๐ธ๐๐ โฃ. Similarly, โฃ๐ธ๐๐ โฃ + 2โฃ๐ธ๐ โฃ = ๐โฃ๐ โฃ. Since, ๐บ[๐ธ๐ ] does not contain the path of order ๐, according to Theorem 3.1, we have โฃ๐ธ๐ โฃ โค โฃ๐ โฃ(๐โ2) . Combining all 2 the previous formulas we immediately have โฃ๐โฃ โฅ
1 1 1 ๐โ๐+2 โฃ๐ธ๐๐ โฃ = (๐โฃ๐ โฃ โ 2โฃ๐ธ๐ โฃ) โฅ (๐โฃ๐ โฃ โ โฃ๐ โฃ(๐ โ 2)) = โฃ๐ โฃ. ๐ ๐ ๐ ๐
Then โฃ๐โฃ + โฃ๐ โฃ โฃ๐ โฃ ๐ 2๐ โ ๐ + 2 =1+ โค1+ = โฃ๐โฃ โฃ๐โฃ ๐โ๐+2 ๐โ๐+2 5
and โฃ๐โฃ โฅ
๐โ๐+2 โฃ๐ (๐บ)โฃ. 2๐ โ ๐ + 2
Corollary 3.1. If ๐บ is a cubic graph then ๐๐ (๐บ) โฅ ๐3 (๐บ) โฅ 25 โฃ๐ (๐บ)โฃ.
5โ๐ 8โ๐ โฃ๐
(๐บ)โฃ, in particular
Corollary 3.2. Let ๐ โฅ 2 be a ๏ฌxed positive integer and ๐๐ the ๐-dimensional hypercube. Then 1 ๐๐ (๐๐ ) = . lim ๐โโ โฃ๐ (๐๐ )โฃ 2 ๐โ๐+2 Proof. By Theorem 3.2 we know that 2๐โ๐+2 โฃ๐ (๐๐ )โฃ โค ๐๐ (๐๐ ) โค ๐2 (๐๐ ) โค 1 2 โฃ๐ (๐๐ )โฃ, and the result immediately follows.
Theorem 3.3. Let ๐ โฅ 2 and ๐ โฅ ๐ โ 1 be positive integers. There exists a ๐-regular graph ๐บ with ๐๐ (๐บ) โค
๐โ๐+2 โฃ๐ (๐บ)โฃ. 2๐ โ ๐ + 2
Proof. Given values of ๐ and ๐, a graph ๐บ may be constructed as follows. The vertex set is ๐ (๐บ) = ๐ด โช ๐ต with โฃ๐ดโฃ = (๐ โ 1)(๐ โ ๐ + 2) โฃ๐ตโฃ = ๐(๐ โ 1). Let the set ๐ด be an independent set in ๐บ and let ๐ต consist of ๐ components of ๐พ๐โ1 in ๐บ. The remaining edges between ๐ด and ๐ต may be de๏ฌned arbitrarily to ๏ฌll ๐-regularity of ๐บ. Clearly, ๐ด is a ๐-path vertex cover of ๐บ since ๐ต does not contain a path on ๐ vertices and โฃ๐ดโฃ (๐ โ 1)(๐ โ ๐ + 2) ๐โ๐+2 ๐๐ (๐บ) โค = = . โฃ๐ (๐บ)โฃ โฃ๐ดโฃ + โฃ๐ตโฃ (๐ โ 1)(๐ โ ๐ + 2) + ๐(๐ โ 1) 2๐ โ ๐ + 2
4
Cartesian products
Recall that the Cartesian product ๐บโก๐ป of graphs ๐บ = (๐ (๐บ), ๐ธ(๐บ)) and ๐ป = (๐ (๐ป), ๐ธ(๐ป)) has the vertex set ๐ (๐บ)ร๐ (๐ป), and vertices (๐ข, ๐ฃ), (๐ฅ, ๐ฆ) are adjacent whenever ๐ข = ๐ฅ and ๐ฃ๐ฆ โ ๐ธ(๐ป), or ๐ข๐ฅ โ ๐ธ(๐บ) and ๐ฃ = ๐ฆ. By ๐๐บ and ๐๐ป we denote the natural projections to the factors ๐บ and ๐ป, respectively. For a ๏ฌxed vertex ๐ข โ ๐ (๐บ), the ๐ป-๏ฌber ๐ข ๐ป is the subgraph of ๐บโก๐ป induced by the set of vertices {(๐ข, ๐ฃ), ๐ฃ โ ๐ (๐ป)}; analogously, for a ๏ฌxed vertex ๐ฃ โ ๐ (๐ป), the ๐บ-๏ฌber ๐บ๐ฃ is the subgraph of ๐บโก๐ป induced by the set of vertices {(๐ข, ๐ฃ), ๐ข โ ๐ (๐บ)}. Note that each ๐ป-๏ฌber is isomorphic to ๐ป, and is an induced subgraph of ๐บโก๐ป. Hence any vertex ๐-path cover ๐ of ๐บโก๐ป must contains at least 6
๐๐ (๐ป) vertices in every ๐ป-๏ฌber. Since there are โฃ๐ (๐บ)โฃ such ๏ฌbers, we get โฃ๐โฃ โฅ โฃ๐ (๐บ)โฃ โ
๐๐ (๐ป). By reversing the roles of ๐บ and ๐ป we immediately get the following result. Proposition 4.1. Let ๐บ and ๐ป be graphs and ๐ โฅ 2 be a positive integer. Then ๐๐ (๐บโก๐ป) โฅ max{๐๐ (๐บ) โ
โฃ๐ (๐ป)โฃ, โฃ๐ (๐บ)โฃ โ
๐๐ (๐ป)}. For ๐ = 2 this bound was observed already by Vizing [16], who expressed it in terms of independence numbers as follows ๐ผ(๐บโก๐ป) โค min{๐ผ(๐บ) โฃ๐ (๐ป)โฃ , ๐ผ(๐ป) โฃ๐ (๐บ)โฃ}.
(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 ๐บ to a graph ๐ป is a mapping โ : ๐ (๐บ) โ ๐ (๐ป) such that for any pair ๐ข, ๐ฃ โ ๐ (๐บ): if ๐ข๐ฃ โ ๐ธ(๐บ) then โ(๐ข)โ(๐ฃ) โ ๐ธ(๐ป). If there exists a homomorphism from a graph ๐บ to a graph ๐ป we write ๐บ โ ๐ป. Let ๐บ be a graph, ๐ โฅ 2 an integer, and let ๐ด,๐ต be two subsets of ๐ (๐บ). We say that ๐ด, ๐ต form a good ๐-partition in ๐บ if there is no path on ๐ vertices in ๐บ of the form ๐1 , . . . , ๐๐1 = ๐๐1 , . . . , ๐๐2 = ๐๐2 , . . . , ๐๐ such that ๐๐ โ ๐ด, ๐๐ โ ๐ต (for all applicable indices ๐ and ๐); in addition, we allow that the path ends with ๐๐ and/or begins with ๐1 . In other words, there is no ๐ path in ๐ด โช ๐ต. In particular, any two ๐๐ -sets ๐ด, ๐ต such that ๐ด โฉ ๐ต = โ
form a good ๐-partition of ๐บ. On the other hand, if one of the sets ๐ด, ๐ต is not a vertex ๐-path cover then ๐ด, ๐ต will clearly not form a good ๐-partition of ๐บ; indeed, if say ๐ด is not a vertex ๐-path cover then there is already a path of length ๐ in ๐ (๐บ) โ ๐ด that begins with ๐1 and ends with ๐๐ . We say that a vertex ๐-path cover of minimum size ๐๐ (๐บ) is a ๐๐ -set of ๐บ, and let ๐ฎ be the set of all ๐๐ -sets of ๐บ. Then the ๐๐ -graph of ๐บ, denoted ฮจ๐ (๐บ) is the graph with ๐ (ฮจ๐ (๐บ)) = ๐ฎ, and ๐ด, ๐ต โ ๐ฎ are adjacent whenever they form a good ๐-partition in ๐บ. Theorem 4.1. Let ๐บ, ๐ป be arbitrary graphs. If ๐๐ (๐บโก๐ป) = max{๐๐ (๐บ) โ
โฃ๐ (๐ป)โฃ, โฃ๐ (๐บ)โฃ โ
๐๐ (๐ป)} then ๐ป โ ฮจ๐ (๐บ) or ๐บ โ ฮจ๐ (๐ป). Proof. Suppose ๏ฌrst that ๐บ and ๐ป are graphs such that ๐๐ (๐บ)โฃ๐ (๐ป)โฃ โฅ ๐๐ (๐ป)โฃ๐ (๐บ)โฃ, and that the equality holds, thus ๐๐ (๐บโก๐ป) = ๐๐ (๐บ)โฃ๐ (๐ป)โฃ. 7
Let ๐ด be a ๐๐ -set of ๐บโก๐ป. Note that in each ๐บ-๏ฌber of ๐บโก๐ป at least ๐๐ (๐บ) vertices are from ๐ด. Combining this with โฃ๐ดโฃ = ๐๐ (๐บ)โฃ๐ (๐ป)โฃ we ๏ฌnd that in each ๐บ-๏ฌber there are exactly ๐๐ (๐บ) vertices from ๐ด. In other words, vertices of ๐ด in each ๐บ-๏ฌber form a ๐๐ -set of this ๐บ-๏ฌber. Let ๐ข, ๐ฃ โ ๐ (๐ป) be adjacent vertices. Then we claim that the sets ๐๐บ (๐ดโฉ๐บ๐ข ) and ๐๐บ (๐ดโฉ๐บ๐ฃ ) are two ๐๐ -sets of that form a good ๐-partition of ๐บ. Indeed, otherwise there would exist a path of length ๐ in ๐บ of the form ๐1 , . . . , ๐๐1 = ๐๐1 , . . . , ๐๐2 = ๐๐2 , . . . , ๐๐ as described above, and hence in ๐บโก๐ป โ๐ด there would be a path (๐1 , ๐ข), . . . , (๐๐1 , ๐ข), (๐๐1 , ๐ฃ), . . . , (๐๐2 , ๐ฃ), (๐๐2 , ๐ข), . . . , (๐๐ , ๐ข) of the same length ๐. This is a contradiction with ๐ด being a vertex ๐path cover of ๐บ. Hence the mapping โ : ๐ป โ ฮจ๐ (๐บ), de๏ฌned by โ(๐ข) = ๐๐บ (๐ด โฉ ๐บ๐ข ) is a homomorphism, since for any ๐ข๐ฃ โ ๐ธ(๐ป), also โ(๐ข) and โ(๐ฃ) 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, ๐ป is isomorphic to the complete graph ๐พ๐ , then ฮจ๐ (๐ป) 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 (๐พ๐ ) is nontrivial when 2๐ โฅ ๐. The part from Theorem 4.1, i.e. ๐ป โ ฮจ๐ (๐พ๐ ), thus translates to the colorings of a graph ๐ป with ๐combinations. The following theorem, containing the formulas for ๐๐ of grids (Cartesian products of paths), is the main result of this section. Theorem 4.2.
(i) ๐3 (๐2๐+1 โก ๐2๐ ) = 2๐๐ + โ 2๐ 3 โ, where ๐, ๐ โฅ 1,
(ii) ๐3 (๐2๐ โก ๐2๐ ) = 2๐๐, where ๐, ๐ โฅ 1, (iii) ๐3 (๐2๐+1 โก ๐2๐+1 ) = ๐(2๐ + 1) + โ 2๐+1 3 โ, where 1 โค ๐ โค ๐. Proof. (i) Let us label the vertices of ๐2๐+1 consecutively by ๐ข1 , ๐ข2 , . . . , ๐ข2๐+1 and the vertices of ๐2๐ by ๐ฃ1 , ๐ฃ2 , . . . , ๐ฃ2๐ . With this labeling, the vertex ๐ฃ๐ (๐ข๐ , ๐ฃ๐ ) โ ๐ (๐2๐+1 โก ๐2๐ ) is in ๐th ๏ฌber ๐ข๐ ๐2๐ and in ๐ th ๏ฌber ๐2๐+1 . 2๐ Let us ๏ฌrst prove that ๐3 (๐2๐+1 โก ๐2๐ ) โค 2๐๐ + โ 3 โ. We construct a vertex 3-path cover set ๐ in the following way: ๐ = {(๐ข2๐+1 , ๐ฃ3๐+3 ), (๐ข2๐ , ๐ฃ3๐+1 ), (๐ข2๐ , ๐ฃ3๐+2 )โฃ for applicable indices ๐ and ๐} . It is not di๏ฌcult 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 ๐2๐+1 โก ๐2๐ Now let us prove that ๐3 (๐2๐+1 โก ๐2๐ ) โฅ 2๐๐ + โ 2๐ 3 โ. Let ๐ be the ๐ข ๐ optimal vertex 3-path covering. Consider the ๏ฌber ๐2๐ , for some ๐ โ {1, 2, . . . , 2๐}. Suppose (๐ข๐ , ๐ฃ๐ ) โโ ๐, for some ๐ โ {1, 2, . . . , 2๐ โ1}, and all its neighbors ๐ข in ๐ ๐2๐ are in ๐. Then in the ๏ฌber ๐ข๐+1 ๐2๐ either the vertex (๐ข๐+1 , ๐ฃ๐ ) is in ๐ (left-hand side of Fig. 2) or all its neighbors in this ๏ฌber are in ๐ (right-hand side of Fig. 2).
vj
vj
ui ui 1
ui ui 1
Figure 2: Vertex (๐ข๐ , ๐ฃ๐ ) is not in ๐ Suppose (๐ข๐ , ๐ฃ๐ ) and (๐ข๐ , ๐ฃ๐+1 ), for some ๐ โ {1, 2, . . . , 2๐ โ 1}, are not in ๐. Then the vertices (๐ข๐+1 , ๐ฃ๐ ) and (๐ข๐+1 , ๐ฃ๐+1 ) must be in ๐, otherwise ๐ is not a vertex 3-path covering (see Fig. 3). Let ๐ด be the set of all vertices in ๏ฌber ๐ข๐ ๐2๐ that are not in ๐, and ๐ต the set of all vertices in ๏ฌber ๐ข๐+1 ๐2๐ that are in ๐. We shall prove that โฃ๐ดโฃ โค โฃ๐ตโฃ by ๏ฌnding a one to one function ๐ from ๐ด to ๐ต. Let (๐ข๐ , ๐ฃ๐ ) be a vertex in ๐ด. Suppose that the vertex (๐ข๐ , ๐ฃ2๐ ) is not in ๐ด. By Fig. 2 and Fig. 3 we set ๐ ((๐ข๐ , ๐ฃ๐ )) = (๐ข๐+1 , ๐ฃ๐ ) if (๐ข๐+1 , ๐ฃ๐ ) is in ๐, otherwise we set ๐ ((๐ข๐ , ๐ฃ๐ )) = (๐ข๐+1 , ๐ฃ๐+1 ). This is obviously a one to one function. We take a di๏ฌerent approach if the vertex (๐ข๐ , ๐ฃ2๐ ) is in ๐ด. If also the vertex (๐ข๐ , ๐ฃ2๐โ1 ) is in ๐ด we can use the same argument then above since 9
vj vj 1
ui ui 1 Figure 3: Vertices (๐ข๐ , ๐ฃ๐ ) and (๐ข๐ , ๐ฃ๐+1 ) are not in ๐ according to Fig. 3 we can set ๐ ((๐ข๐ , ๐ฃ2๐ )) = (๐ข๐+1 , ๐ฃ2๐ ). If the vertex (๐ข๐ , ๐ฃ2๐โ1 ) โ / ๐ด and (๐ข๐+1 , ๐ฃ2๐ ) โ ๐ต then we can set ๐ ((๐ข๐ , ๐ฃ2๐ )) = (๐ข๐+1 , ๐ฃ2๐ ) otherwise we can set ๐ ((๐ข๐ , ๐ฃ2๐ )) = (๐ข๐+1 , ๐ฃ2๐โ1 ). It can happen that the vertex (๐ข๐+1 , ๐ฃ2๐โ1 ) is already an image of the vertex (๐ข๐ , ๐ฃ2๐โ2 ). If this situation occurs we can set ether ๐ ((๐ข๐ , ๐ฃ2๐โ2 )) = (๐ข๐+1 , ๐ฃ2๐โ2 ) or ๐ ((๐ข๐ , ๐ฃ2๐โ2 )) = (๐ข๐+1 , ๐ฃ2๐โ3 ) since at least one of the vertices (๐ข๐+1 , ๐ฃ2๐โ2 ) or (๐ข๐+1 , ๐ฃ2๐โ3 ) is in ๐ต. Note that if vertex (๐ข๐ , ๐ฃ2๐โ2 ) exists in the grid then so must vertex (๐ข๐+1 , ๐ฃ2๐โ3 ) since every ๐2๐ -๏ฌber has even number of vertices. It can occur that the vertex (๐ข๐+1 , ๐ฃ2๐โ3 ) is also an image of a vertex in ๏ฌber ๐ข๐ ๐2๐ . In this case we can repeat the above procedure. We can always ๏ฌnd a one to one function from ๐ด to ๐ต and hence โฃ๐ดโฃ โค โฃ๐ตโฃ. Therefore two consecutive ๏ฌbers, ๐ข๐ ๐2๐ and ๐ข๐+1 ๐2๐ , must have at least 2๐ vertices in ๐. There are ๐ such paired ๏ฌbers in ๐2๐+1 โก ๐2๐ . Together with the additional ๏ฌber, which is isomorphic to the path ๐2๐ , there must be at lest 2๐๐ + โ 2๐ 3 โ 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 ๏ฌber.
(iii) Note that in the proof of (i), two consecutive ๏ฌbers, ๐ข๐ ๐2๐ and ๐ข๐+1 ๐ , have at least 2๐ vertices in ๐. The ๏ฌber ๐ข๐ ๐ 2๐ 2๐+1 in this case has odd number of vertices and it can occur that two consecutive ๏ฌbers do not have at least 2๐ + 1 vertices in ๐. Nevertheless, two consecutive ๏ฌbers, ๐ข๐ ๐ ๐ข๐+1 ๐ 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 ๏ฌbers ๐ข๐ ๐2๐+1 and ๐ข๐+1 ๐2๐+1 the vertices (๐ข๐ , ๐ฃ2๐ ) and (๐ข๐+1 , ๐ฃ2๐ ), ๐ โ {1, . . . , ๐}, must all be in ๐ (see the structure of sets ๐ด and ๐ต in (i)). Hence, both ๏ฌbers ๐ข๐ ๐2๐+1 and ๐ข๐+1 ๐2๐+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๏ฌces. Now we want to prove that ๐3 (๐2๐+1 โก ๐2๐+1 ) โฅ ๐(2๐ + 1) + โ 2๐+1 3 โ. Asumme that 0 โค ๐ โค ๐ and ๐ is the optimal vertex 3-path covering of the graph ๐2๐+1 โก ๐2๐+1 . We proceed with induction on ๐. For ๐ = 0, the graph ๐1 โก ๐2๐+1 is isomorphic to the path ๐2๐+1 , and hence ๐3 (๐2๐+1 ) = โ 2๐+1 3 โ. In the induction step we 10
consider two cases. Suppose that the last two ๏ฌbers, ๐ข2๐ ๐2๐+1 and ๐ข2๐+1 ๐2๐+1 , contribute at least 2๐ + 1 vertices to ๐. Assume that ๐3 (๐2๐โ1 โก ๐2๐+1 ) = (๐ โ 1)(2๐ + 2๐+1 1) + โ 2๐+1 3 โ and ๐3 (๐2๐+1 โก ๐2๐+1 ) < ๐(2๐ + 1) + โ 3 โ. If we remove the ๐ข ๐ข ๏ฌbers, 2๐ ๐2๐+1 and 2๐+1 ๐2๐+1 , we remove at least 2๐ + 1 vertices from ๐ an therefore ๐3 (๐2๐โ1 โก ๐2๐+1 ) < ๐(2๐ + 1) + โ 2๐+1 3 โ โ 2๐ โ 1. By induction assumption it follows that โ โ โ โ 2๐ + 1 2๐ + 1 (๐ โ 1)(2๐ + 1) + < ๐(2๐ + 1) + โ 2๐ โ 1 . 3 3 By doing some basic calculation this leads us to a contradiction. For the second case assume that the last two ๏ฌbers, ๐ข2๐ ๐2๐+1 and ๐ข2๐+1 ๐2๐+1 , contribute exactly 2๐ vertices to ๐. According to the observation above, if we remove the last ๏ฌber ๐ข2๐+1 ๐2๐+1 from the graph ๐2๐+1 โก ๐2๐+1 we remove at least ๐ vertices from ๐. Suppose again that ๐3 (๐2๐+1 โก ๐2๐+1 ) < 2๐+1 ๐(2๐ + 1) + โ 2๐+1 3 โ. Then ๐3 (๐2๐ โก ๐2๐+1 ) < ๐(2๐ + 1) + โ 3 โ โ ๐. But 2๐ according to (i) we know that ๐3 (๐2๐ โก ๐2๐+1 ) = 2๐๐ + โ 3 โ and hence โ โ โ โ 2๐ 2๐ + 1 2๐๐ + < ๐(2๐ + 1) + โ๐. 3 3 By doing some basic calculation we get that ๐ < ๐, which again is a contradiction since ๐ โค ๐. As seen in Theorem 4.2, it is already hard to determine the exact value of ๐3 for grids therefore it would be nice to have at least some lower or upper bounds for ๐๐ , ๐ โฅ 4. Lemma 4.1. For each ๐ โฅ 2, ๐๐2 (๐2๐ โก ๐3๐ ) โฅ ๐. Proof. Assume to the contrary that ๐ is a ๐-path vertex cover of ๐2๐ โก ๐3๐ , with โฃ๐โฃ โค ๐ โ 1. Then ๐2๐ โก ๐3๐ has at least ๐ ๐3๐ -layers not containing any vertex of ๐. Denote such layers by ๐๐1 , ๐๐2 , . . . , ๐๐๐ . Since โฃ๐โฃ โค ๐ โ 1, then two consecutive layers can be connected vertically in ๏ฌrst and last thirds without crossing ๐. Formally, for each ๐ โ {1, . . . , (๐ โ 1)} there exists ๐๐ โ {1, . . . , ๐} and ๐๐ โ {2๐ + 1, . . . , 3๐} such that for each โ โ {๐๐ , . . . , ๐๐+1 } the vertices (๐๐ , โ), (๐๐ , โ) โ / ๐. Finally, we construct a path ๐ moving horizontally on ๐๐1 , ๐๐2 , . . . , ๐๐๐ layers and vertically on ๐๐ , ๐๐ columns: ๐ =(1, ๐1 ), . . . , (1๐ , ๐1 ), (1๐ , ๐2 ), . . . , (2๐ , ๐2 ), (2๐ , ๐3 ), . . . , (3๐ , ๐3 ), (3๐ , ๐4 ), . . . , (4๐ , ๐4 ), . . . . Such a path contains at least ๐ vertices in each layer ๐๐1 , ๐๐2 , . . . , ๐๐๐ , hence the total order of the path ๐ is at least ๐2 , a contradiction. 11
First we present a lower bound with the help of Lemma 4.1. โ โ Proposition 4.2. For ๐ โฅ 4, ๐ โฅ 2 ๐, ๐ โฅ 3 ๐, the following holds ๐๐ โ โค ๐๐ (๐๐ โก ๐๐ ). 24 ๐ Proof. We split the whole graph ๐๐ โ โก ๐๐ โ into ๐ disjoint subgraphs isomorphic to ๐2โ๐ โก ๐3โ๐ such that ๐(2 ๐)(3 ๐) โฅ 41 ๐๐. By Lemma 4.1 a โ ๐-path vertex cover must have at least ๐ vertices in each subgraph isomorphic to ๐2โ๐ โก ๐3โ๐ in ๐บ, hence: โ ๐๐ ๐๐ (๐๐ โก ๐๐ ) โฅ ๐ ๐ โฅ โ . 24 ๐
We conclude the paper with the following upper bound. Proposition 4.3. For ๐ โฅ 4 the following holds 2๐๐ ๐๐ . ๐๐ (๐๐ โก ๐๐ ) โค โ โ ๐ ๐ โ Proof. We will construct a ๐-path vertex cover with at most 2๐๐ โ ๐๐ ๐ ๐ { } โ vertices. Let ๐1 = (๐, ๐) โ ๐๐ โก ๐๐ โฃ ๐ โก 0 (modโ ๐โ) (for all applicable { โ } indicies ๐ and ๐) and similarly ๐2 = (๐, ๐) โ ๐๐ โก ๐๐ โฃ ๐ โก 0 (modโ ๐โ) . It is easy to see that ๐ = (๐1 โช ๐2 )โ(๐1 โฉ ๐2 ) is a ๐-path vertex cover, since the largest subgraph of ๐๐ โก ๐๐ with all vertices uncovered is isomorphic to ๐โโ๐โโ1 โก ๐โโ๐โโ1 . โ In a ๐๐ -layer we cover each โ ๐โ-th vertex, since there ๐ layers the โ . Similarly, โฃ๐2 โฃ โค ๐๐ โ . The vertices size of ๐1 is at most โฃ๐1 โฃ โค โ๐๐ ๐โ โ ๐โ
(๐, ๐) โ ๐1 โฉ ๐2 can be left uncovered, since all the vertices (๐ ยฑ 1, ๐) and (๐, ๐ ยฑ 1) are in ๐. Since the size of โฃ๐1 โฉ ๐2 โฃ โค ๐๐ 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) 219โ236. [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) 241โ253.
12
[3] M. Borowiecki, I. Broere, M. Frick, P. Mihยดok and G. Semaniหsin, Survey of hereditary properties of graphs, Discuss. Math. - Graph Theory 17 (1997) 5โ50. [4] B. Breหsar, F. Kardoหs, J. Katreniหc, G. Semaniหsin, Minimum ๐-path vertex cover, Discrete Appl. Math. 159 (12) (2011) 1189โ1195. 1189โ1195. [5] B. Breหsar and B. Zmazek, On the independence graph of a graph, Discrete Math. 272 (2001), 263โ268. [6] K. Cameron, P. Hell, Independent packings in structured graphs, Math. Program. 105 (2-3) (2006) 201โ213. [7] P. Erdหos and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959) 337โ356. [8] F. Gยจoring, J. Harant, D. Rautenbach and I. Schiermeyer, On Findependence in graphs, Discuss. Math. - Graph Theory 29 (2) (2009) 377โ383. [9] P. Hell, X. Yu, H. Zhou, Independence ratios of graph powers, Discrete Math. 127 (1994) 213โ220. [10] F. Kardoหs, J. Katreniหc, 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) 237โ238. [12] V. V. Lozin, D. Rautenbach, Some results on graphs without long induced paths, Inf. Process. Lett. 88 (4) (2003) 167โ171. [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) 1352โ1366. [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), 30โ43. 13
[17] M. Yannakakis, Node-deletion problems on bipartite graphs, SIAM J. Computing 10 (1981) 310โ327.
14