International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 193560 | 10 pages | https://doi.org/10.5402/2011/193560

Graph Polynomials

Academic Editor: M. Przybylska
Received23 May 2011
Accepted05 Jul 2011
Published25 Aug 2011

Abstract

One of the most important and applied concepts in graph theory is to find the edge cover, vertex cover, and dominating sets with minimum cardinal also to find independence and matching sets with maximum cardinal and their polynomials. Although there exist some algorithms for finding some of them (Kuhn and Wattenhofer, 2003; and Mihelic and Robic, 2005), but in this paper we want to study all of these concepts from viewpoint linear and binary programming and we compute the coefficients of the polynomials by solving a system of linear equations with {0,1} variables.

1. Introduction

All graphs in this note are simple, connected, finite, and undirected, though it is probable that some of the obtained results are extendable to general or directed graphs.

Let 𝐺=(𝑉,𝐸) be a simple and connected graph with |𝑉|=𝑛 and |𝐸|=𝑚; then the edge cover and edge dominating polynomials are of degree 𝑚, and the vertex cover and dominating polynomials are of degree 𝑛, in which coefficient of 𝑥𝑘 is the number of edge cover, edge dominating, vertex cover, and dominating sets with 𝑘 elements, respectively. Also the independence and matching polynomials are at most of degree 𝑛 such that coefficient of 𝑥𝑘 is the number of independence and matching sets with 𝑘 elements, respectively, for some positive integer 𝑘.

For some notation being not defined here we refer the reader to [1].

A set 𝐿⊆𝐸 is an edge cover if every vertex 𝑣∈𝑉 is incident to some edge of 𝐿.

A set 𝑄⊆𝑉 is a vertex cover if every edge 𝑒∈𝐸 has at least one endpoint in 𝑄.

A set 𝑆⊆𝑉 is an independent set if non of two vertex in 𝑆 are not adjacent.

The maximum size of an independent set is named independence number.

A matching in graph 𝐺 is a set 𝑀⊆𝐸 with no shared endpoints.

In graph 𝐺 a set 𝐷⊆𝑉 is a dominating set if every vertex number in 𝐷 has a neighbor in 𝐷, and finally a set 𝑊⊆𝐸 is an edge dominating set if every edge number in 𝑊 has a neighbor in 𝑊.

We set ||𝐿||Min=ğ›½î…ž,||𝑆||||𝑄||||𝑀||Max=𝛼,Min=𝛽,Max=ğ›¼î…ž,||𝐷||||𝑊||Min=𝛾,Min=ğ›¾î…ž.(1.1) By [1],(i)𝛼+𝛽=𝑛, (ii)ğ›¼î…ž+ğ›½î…ž=𝑛.

Also in every bipartite graph(iii)𝛼=ğ›½î…ž, (iv)ğ›¼î…ž=𝛽.

We denote the Adjacency matrix by 𝐴 and Incidence matrix by 𝑅, in which 𝐴=[ğ‘Žğ‘–ğ‘—]𝑛×𝑛 such that ğ‘Žğ‘–ğ‘—=thenumbersofedgeswithendpoints𝑣𝑖and𝑣𝑗,(1.2) and also 𝑅=[𝑟𝑖𝑗]𝑛×𝑚 in which 𝑟𝑖𝑗=1,𝑣𝑖isanendpointof𝑒𝑗,0,otherwise.(1.3) We also define an Edge adjacency matrix 𝐵=[𝑏𝑖𝑗]𝑚×𝑚 as follows: 𝑏𝑖𝑗=1,𝑒𝑖isadjacentto𝑒𝑗,0,otherwise,(1.4) and 𝑏𝑖𝑖=0.

From now on we set 𝑣𝑉=1,𝑣2,…,𝑣𝑛𝑡,𝑒𝐸=1,𝑒2,…,𝑒𝑚𝑡,1𝑛=(1,1,…,1)𝑡1×𝑛.(1.5)

2. Edge Cover Set and Edge Cover Polynomial

As previous notations we have the following theorem for obtaining the minimum size of edge cover set.

Theorem 2.1. One has 𝛽′=min𝑚𝑖=1𝑒𝑖subjectto𝑅𝐸≧1𝑛,𝑒𝑖∈{0,1},where𝑖=1,2,…,𝑚.(2.1)

Proof. Since an edge cover set of 𝐺 is a set 𝐿 of edges such that every vertex of 𝐺 is incident to some edge of 𝐿 and we want to obtain the optimal size of the sets in covering problems, so we will have a minimize problem; that is, the object function is ğ›½î…žâˆ‘=min𝑚𝑖=1𝑒𝑖; on the other hand for each 𝑣𝑖∈𝑉 at least one edge with endpoint 𝑣𝑖 must belong to 𝐿; in other words from every row of matrix 𝑅 at least one entry (𝑒𝑖) must be equal to 1. Therefore 𝑟11𝑒1+𝑟12𝑒2+⋯+𝑟1𝑚𝑒𝑚𝑟≥1,21𝑒1+𝑟22𝑒2+⋯+𝑟2𝑚𝑒𝑚⋮𝑟≥1,𝑛1𝑒1+𝑟𝑛2𝑒2+⋯+𝑟𝑛𝑚𝑒𝑚𝑒≥1,𝑖∈{0,1},where𝑖=1,2,…,𝑚.(2.2)

Definition 2.2. An edge cover polynomial is as follows: 𝐿(𝑥)=ğ‘Ž0𝑥𝛽′+ğ‘Ž1𝑥𝛽′+1+⋯+ğ‘Žğ‘šâˆ’ğ›½â€²ğ‘¥ğ‘š,(2.3) where ğ›½î…ž is the same as in (2.1) and ğ‘Žğ‘–â€™s are the number of edge cover sets with ğ›½î…ž+𝑖 elements.

Theorem 2.3. The coefficients ğ‘Ž0,ğ‘Ž1,…,ğ‘Žğ‘šâˆ’ğ›½â€² in edge cover polynomial are all of solutions of the following system for 𝑖=0,𝑖=1,…,𝑖=ğ‘šâˆ’ğ›½î…ž, respectively, 𝑅𝐸≥1𝑛,𝑒(∗)1+𝑒2+⋯+𝑒𝑚=ğ›½î…žğ‘’+𝑖,𝑗∈{0,1},ğ‘¤â„Žğ‘’ğ‘Ÿğ‘’ğ‘—=1,2,…,𝑚.(∗∗)

Proof. The first inequality ∗ is the condition for a set to be an edge cover set and ∗∗ for each 𝑖 causes that we have the edge cover sets with cardinality ğ›½î…ž,ğ›½î…ž+1,…,𝑚, respectively, and with this process we can compute ğ‘Ž0,ğ‘Ž1,…,ğ‘Žğ‘šâˆ’ğ›½â€². It is trivial that ğ‘Žğ‘šâˆ’ğ›½â€²=1 and this completes the proof.

Algorithm 2.4 (For computation ğ‘Žğ‘–). One has the following.Step 1. Solve 𝛽′=min𝑚𝑖=1𝑒𝑖,𝑅𝐸≥1𝑛,𝑒𝑗∈{0,1},where𝑗=1,2,…,𝑚,(2.4) and obtain ğ›½î…ž.Step 2. For 𝑖=0 to ğ‘šâˆ’ğ›½î…žâˆ’1, compute all of solutions: 𝑅𝐸≥1𝑛,𝑒1+𝑒2+⋯+𝑒𝑚=ğ›½î…žğ‘’+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑚.(2.5)Step 3. Set ğ‘Žğ‘– to be equal to all solutions of Step 2.

3. Independence Set and Independence Polynomial

In an independence set 𝑆 from every two adjacent vertices at most one of them belongs to 𝑆; this means that for all 𝑒𝑖∈𝐸 with end points 𝑣𝑖 and 𝑣𝑗 at most 𝑣𝑖 or 𝑣𝑗 belongs to 𝑆. Therefore we have the following.

Theorem 3.1. One has 𝛼=max𝑛𝑖=1𝑣𝑖subjectto𝑅𝑡𝑉≤1𝑚,𝑣𝑖∈{0,1},where𝑖=1,2,…,𝑛.(3.1)

Definition 3.2. An independence polynomial is as follows: 𝑆(𝑥)=𝑏0𝑥𝛼+𝑏1𝑥𝛼−1+⋯+𝑏𝛼−1𝑥,(3.2) where 𝛼 is the same as in (3.1) and 𝑏𝑖’s are the numbers of independence sets with 𝛼−𝑖 elements.

Theorem 3.3. The coefficients 𝑏0,𝑏1,…,𝑏𝛼−1 are all of solutions of the following system 𝑖=0,𝑖=1,…,𝑖=𝛼−1, respectively, 𝑅𝑡𝑉≤1𝑚,𝑣1+𝑣2+⋯+𝑣𝑛𝑣=𝛼−𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(3.3)

Algorithm 3.4 (For computation 𝑏𝑖). One has the following.Step 1. Solve 𝛼=max𝑛𝑖=1𝑣𝑖,𝑅𝑡𝑉≤1𝑚,𝑣𝑗∈{0,1},where𝑗=1,2,…,𝑛,(3.4) and obtain 𝛼.Step 2. For 𝑖=0 to 𝛼−2, compute all of solutions: 𝑅𝑡𝑉≤1𝑚,𝑣1+𝑣2+⋯+𝑣𝑛𝑣=𝛼−𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(3.5)Step 3. Set 𝑏𝑖 to be equal to all solutions of Step 2 of course 𝑏𝛼−1=𝑛.

4. Vertex Cover Set and Vertex Cover Polynomial

We have the following theorem for vertex cover set.

Theorem 4.1. One has 𝛽=min𝑛𝑖=1𝑣𝑖subjectto𝑅𝑡𝑉≧1𝑚,𝑣𝑖∈{0,1},where𝑖=1,2,…,𝑛.(4.1)

Proof. Since a vertex cover set of 𝐺 is a set 𝑄 of vertices such that every edge of 𝐺 is incident to some vertex of 𝑄 and we want to obtain the optimal size of the sets in covering problems, so we will have a minimize problem; that is, the object function is ∑𝛽=min𝑛𝑖=1𝑣𝑖; on the other hand for each 𝑒𝑖∈𝐸 with endpoint 𝑣𝑖 and 𝑣𝑗 at least one of them must belong to 𝑄; in other words from every row of matrix 𝑅𝑡 at least one entry (𝑣𝑖) must be equal to 1. Therefore 𝑟11𝑣1+𝑟21𝑣2+⋯+𝑟𝑛1𝑣𝑛𝑟≥1,12𝑣1+𝑟22𝑣2+⋯+𝑟𝑛2𝑣𝑛⋮𝑟≥1,1𝑛𝑣1+𝑟2𝑛𝑣2+⋯+𝑟𝑚𝑛𝑣𝑛𝑣≥1,𝑖∈{0,1},ğ‘¤â„Žğ‘’ğ‘Ÿğ‘’ğ‘–=1,2,…,𝑛.(4.2)

Definition 4.2. A vertex cover polynomial is as follows: 𝑄(𝑥)=𝑐0𝑥𝛽+𝑐1𝑥𝛽+1+⋯+𝑐𝑛−𝛽𝑥𝑛,(4.3) where 𝛽 is the same as in (4.1) and 𝑐𝑖’s are the number of vertex cover sets with 𝛽+𝑖 elements.

Theorem 4.3. The coefficients 𝑐0,𝑐1,…,𝑐𝑛−𝛽 are all of solutions of the following system for 𝑖=0,𝑖=1,…,𝑖=𝑛−𝛽, respectively, 𝑅𝑡𝑣𝑉≥1,(𝚤)1+𝑣2+⋯+𝑣𝑛𝑣=𝛽+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(𝚤𝚤)

Proof. The first inequality 𝚤 is the condition for a set to be a vertex cover set and 𝚤𝚤 for each 𝑖 causes that we have the vertex cover sets with cardinality 𝛽,𝛽+1,…,𝑛, respectively, and with this process we can compute 𝑐0,𝑐1,…,𝑐𝑛−𝛽. It is trivial that 𝑐𝑛−𝛽=1 and this completes the proof.

Algorithm 4.4 (For computation 𝑐𝑖). One has the following.Step 1. Solve 𝛽=min𝑛𝑖=1𝑣𝑖,𝑅𝑡𝑉≥1𝑚,𝑣𝑖∈{0,1},where𝑖=1,2,…,𝑛,(4.4) and obtain 𝛽.Step 2. For 𝑖=0 to 𝑛−𝛽−1, compute all of solutions: 𝑅𝑡𝑉≥1𝑚,𝑣1+𝑣2+⋯+𝑣𝑛𝑣=𝛽+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(4.5)Step 3. Set 𝑐𝑖 to be equal to all solutions of Step 2.

5. Matching Set and Matching Polynomial

In a matching set (𝑀) from every two adjacent edges at most one of them belongs to 𝑀 and this means that for all 𝑒𝑖,𝑒𝑗∈𝐸 with common endpoint 𝑣𝑖 at most 𝑒𝑖 or 𝑒𝑗 belongs to 𝑀. Therefore we have the following.

Theorem 5.1. One has ğ›¼î…ž=max𝑚𝑖=1𝑒𝑖subjectto𝑅𝐸≤1𝑛,𝑒𝑗∈{0,1},where𝑗=1,2,…,𝑚.(5.1)

Definition 5.2. A matching polynomial is as follows: 𝑀(𝑥)=𝑑0𝑥𝛼′+𝑑1𝑥𝛼′−1+⋯+𝑑𝛼′−1𝑥,(5.2) where ğ›¼î…ž is the same as in (5.1) and 𝑑𝑖’s are the number of matching sets with ğ›¼î…žâˆ’ğ‘– elements.

Theorem 5.3. The coefficients 𝑑0,𝑑1,…,𝑑𝛼′−1 are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,…,𝑖=ğ›¼î…žâˆ’1, 𝑅𝐸≤1𝑛,𝑒1+𝑒2+⋯+𝑒𝑚=ğ›¼î…žğ‘’âˆ’ğ‘–,𝑗∈{0,1},where𝑗=1,2,…,𝑚.(5.3)

Algorithm 5.4 (For computation 𝑑𝑖). One has the following.Step 1. Solve ğ›¼î…ž=max𝑚𝑖=1𝑒𝑖,𝑅𝐸≤1𝑛,𝑒𝑗∈{0,1},where𝑗=1,2,…,𝑚,(5.4) and obtain ğ›¼î…ž.Step 2. For 𝑖=0 to ğ›¼î…žâˆ’2, compute all of solutions: 𝑅𝐸≤1𝑛,𝑒1+𝑒2+⋯+𝑒𝑚=ğ›¼î…žğ‘’âˆ’ğ‘–,𝑗∈{0,1},where𝑗=1,2,…,𝑚.(5.5)Step 3. Set 𝑑𝑖 to be equal to all solutions of Step 2, 𝑖=0,1,…,ğ›¼î…žâˆ’2, of course 𝑑𝛼′−1=1.

6. Dominating Set and Dominating Polynomial

With the same argument in the previous sections we have the following theorem.

Theorem 6.1. One has 𝛾=min𝑛𝑖=1𝑣𝑖,subjectto𝐴+𝐼𝑛𝑉≧1𝑛,𝑣𝑖∈{0,1},where𝑖=1,2,…,𝑛.(6.1)

Definition 6.2. A dominating polynomial is as follows: 𝐷(𝑥)=𝑓0𝑥𝛾+𝑓1𝑥𝛾+1+⋯+𝑓𝑛−𝛾𝑥𝑛,(6.2) where 𝛾 is the same as in (6.1) and 𝑓𝑖’s are the number of dominating sets with 𝛾+𝑖 elements.

Theorem 6.3. The coefficients 𝑓0,𝑓1,…,𝑓𝑛−𝛾 are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,…,𝑖=𝑛−𝛾, 𝐴+𝐼𝑛𝑉≥1𝑛𝑣,(†)1+𝑣2+⋯+𝑣𝑛𝑣=𝛾+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(††)

Proof. The first inequality † is the condition for a set to be a dominating set and †† for each 𝑖 causes that we have the dominating sets with cardinality 𝛾,𝛾+1,…,𝑛, respectively, and with this process we can compute 𝑓0,𝑓1,…,𝑓𝑛−𝛾. It is trivial that 𝑓𝑛−𝛾=1 and this completes the proof.

Algorithm 6.4 (For computation 𝑓𝑖). One has the following.Step 1. Solve 𝛾=min𝑛𝑖=1𝑣𝑖,𝐴+𝐼𝑛𝑉≥1𝑛,𝑣𝑖∈{0,1},where𝑖=1,2,…,𝑛,(6.3) and obtain 𝛾.Step 2. For 𝑖=0 to 𝑛−𝛾−1, compute all of solutions: 𝐴+𝐼𝑛𝑉≥1𝑛,𝑣1+𝑣2+⋯+𝑣𝑛𝑣=𝛾+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑛.(6.4)Step 3. Set 𝑓𝑖 to be equal to all solutions of Step 2.

7. Edge Dominating Set and Edge Dominating Polynomial

With the same argument in previous sections we have Theorems 7.1 and 7.3.

Theorem 7.1. One has ğ›¾î…ž=min𝑚𝑖=1𝑒𝑖subjectto𝐵+𝐼𝑚𝐸≧1𝑚,𝑒𝑖∈{0,1},where𝑖=1,2,…,𝑚.(7.1)

Definition 7.2. An edge dominating polynomial is a polynomial such as 𝑊(𝑥)=ğ‘“î…ž0𝑥𝛾′+ğ‘“î…ž1𝑥𝛾′+1+⋯+ğ‘“î…žğ‘šâˆ’ğ›¾â€²ğ‘¥ğ‘š,(7.2) where ğ›¾î…ž is the same as in (7.1) and ğ‘“î…žğ‘–â€™s are the number of edge dominating sets with ğ›¾î…ž+𝑖 elements.

Theorem 7.3. The coefficients ğ‘“î…ž0,ğ‘“î…ž1,…,ğ‘“î…žğ‘šâˆ’ğ›¾â€² are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,…,𝑖=ğ‘šâˆ’ğ›¾î…ž, 𝐵+𝐼𝑚𝐸≥1𝑚,𝑒1+𝑒2+⋯+𝑒𝑚=ğ›¾î…žğ‘’+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑚.(7.3)

Algorithm 7.4 (For computation 𝑓𝑖). One has the following.Step 1. Solve ğ›¾î…ž=min𝑚𝑖=1𝑒𝑖,𝐵+𝐼𝑚𝐸≥1𝑚,𝑒𝑖∈{0,1},where𝑖=1,2,…,𝑚,(7.4) and obtain ğ›¾î…ž.Step 2. For 𝑖=0 to ğ‘šâˆ’ğ›¾î…žâˆ’1, compute all of solutions: 𝐵+𝐼𝑚𝐸≥1𝑚,𝑒1+𝑒2+⋯+𝑒𝑚=ğ›¾î…žğ‘’+𝑖,𝑗∈{0,1},where𝑗=1,2,…,𝑚.(7.5)Step 3. Set ğ‘“î…žğ‘– to be equal to all solutions of Step 2.

References

  1. D. B. West, Introduction to Graph Theory, Prentice Hall Inc., Upper Saddle River, NJ, USA, 1996.
  2. F. Kuhn and R. Wattenhofer, “Distributed combinatorial optimization,” Tech. Rep. 426, Department of computer Science, ETH Zurich, Zurich, Switzerland, 2003. View at: Google Scholar
  3. J. Mihelic and B. Robic, “Solving the k-center problem efficiently with a dominating set,” Algorithm Journal of Computing and Information Technology, vol. 13, no. 3, pp. 225–233, 2005. View at: Google Scholar

Copyright © 2011 Mehdi Alaeiyan and Saeid Mohammadian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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