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ISRN Algebra
Volume 2011 (2011), Article ID 193560, 10 pages
http://dx.doi.org/10.5402/2011/193560
Research Article

Graph Polynomials

Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran

Received 23 May 2011; Accepted 5 July 2011

Academic Editor: M. Przybylska

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.

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 𝐿𝐸 is an edge cover if every vertex 𝑣𝑉 is incident to some edge of 𝐿.

A set 𝑄𝑉 is a vertex cover if every edge 𝑒𝐸 has at least one endpoint in 𝑄.

A set 𝑆𝑉 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 𝑀𝐸 with no shared endpoints.

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

We set ||𝐿||Min=𝛽,||𝑆||||𝑄||||𝑀||Max=𝛼,Min=𝛽,Max=𝛼,||𝐷||||𝑊||Min=𝛾,Min=𝛾.(1.1) By [1],(i)𝛼+𝛽=𝑛, (ii)𝛼+𝛽=𝑛.

Also in every bipartite graph(iii)𝛼=𝛽, (iv)𝛼=𝛽.

We denote the Adjacency matrix by 𝐴 and Incidence matrix by 𝑅, in which 𝐴=[𝑎𝑖𝑗]𝑛×𝑛 such that 𝑎𝑖𝑗=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,,𝑣𝑛𝑡,𝑒𝐸=1,𝑒2,,𝑒𝑚𝑡,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=min𝑚𝑖=1𝑒𝑖subjectto𝑅𝐸1𝑛,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 𝛽=min𝑚𝑖=1𝑒𝑖; on the other hand for each 𝑣𝑖𝑉 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++𝑟1𝑚𝑒𝑚𝑟1,21𝑒1+𝑟22𝑒2++𝑟2𝑚𝑒𝑚𝑟1,𝑛1𝑒1+𝑟𝑛2𝑒2++𝑟𝑛𝑚𝑒𝑚𝑒1,𝑖{0,1},where𝑖=1,2,,𝑚.(2.2)

Definition 2.2. An edge cover polynomial is as follows: 𝐿(𝑥)=𝑎0𝑥𝛽+𝑎1𝑥𝛽+1++𝑎𝑚𝛽𝑥𝑚,(2.3) where 𝛽 is the same as in (2.1) and 𝑎𝑖’s are the number of edge cover sets with 𝛽+𝑖 elements.

Theorem 2.3. The coefficients 𝑎0,𝑎1,,𝑎𝑚𝛽 in edge cover polynomial are all of solutions of the following system for 𝑖=0,𝑖=1,,𝑖=𝑚𝛽, respectively, 𝑅𝐸1𝑛,𝑒()1+𝑒2++𝑒𝑚=𝛽𝑒+𝑖,𝑗{0,1},𝑤𝑒𝑟𝑒𝑗=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 𝛽,𝛽+1,,𝑚, respectively, and with this process we can compute 𝑎0,𝑎1,,𝑎𝑚𝛽. It is trivial that 𝑎𝑚𝛽=1 and this completes the proof.

Algorithm 2.4 (For computation 𝑎𝑖). One has the following.Step 1. Solve 𝛽=min𝑚𝑖=1𝑒𝑖,𝑅𝐸1𝑛,𝑒𝑗{0,1},where𝑗=1,2,,𝑚,(2.4) and obtain 𝛽.Step 2. For 𝑖=0 to 𝑚𝛽1, compute all of solutions: 𝑅𝐸1𝑛,𝑒1+𝑒2++𝑒𝑚=𝛽𝑒+𝑖,𝑗{0,1},where𝑗=1,2,,𝑚.(2.5)Step 3. Set 𝑎𝑖 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 𝑒𝑖𝐸 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=max𝑛𝑖=1𝑣𝑖subjectto𝑅𝑡𝑉1𝑚,𝑣𝑖{0,1},where𝑖=1,2,,𝑛.(3.1)

Definition 3.2. An independence polynomial is as follows: 𝑆(𝑥)=𝑏0𝑥𝛼+𝑏1𝑥𝛼1++𝑏𝛼1𝑥,(3.2) where 𝛼 is the same as in (3.1) and 𝑏𝑖’s are the numbers of independence sets with 𝛼𝑖 elements.

Theorem 3.3. The coefficients 𝑏0,𝑏1,,𝑏𝛼1 are all of solutions of the following system 𝑖=0,𝑖=1,,𝑖=𝛼1, respectively, 𝑅𝑡𝑉1𝑚,𝑣1+𝑣2++𝑣𝑛𝑣=𝛼𝑖,𝑗{0,1},where𝑗=1,2,,𝑛.(3.3)

Algorithm 3.4 (For computation 𝑏𝑖). One has the following.Step 1. Solve 𝛼=max𝑛𝑖=1𝑣𝑖,𝑅𝑡𝑉1𝑚,𝑣𝑗{0,1},where𝑗=1,2,,𝑛,(3.4) and obtain 𝛼.Step 2. For 𝑖=0 to 𝛼2, compute all of solutions: 𝑅𝑡𝑉1𝑚,𝑣1+𝑣2++𝑣𝑛𝑣=𝛼𝑖,𝑗{0,1},where𝑗=1,2,,𝑛.(3.5)Step 3. Set 𝑏𝑖 to be equal to all solutions of Step 2 of course 𝑏𝛼1=𝑛.

4. Vertex Cover Set and Vertex Cover Polynomial

We have the following theorem for vertex cover set.

Theorem 4.1. One has 𝛽=min𝑛𝑖=1𝑣𝑖subjectto𝑅𝑡𝑉1𝑚,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 𝑒𝑖𝐸 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++𝑟𝑛1𝑣𝑛𝑟1,12𝑣1+𝑟22𝑣2++𝑟𝑛2𝑣𝑛𝑟1,1𝑛𝑣1+𝑟2𝑛𝑣2++𝑟𝑚𝑛𝑣𝑛𝑣1,𝑖{0,1},𝑤𝑒𝑟𝑒𝑖=1,2,,𝑛.(4.2)

Definition 4.2. A vertex cover polynomial is as follows: 𝑄(𝑥)=𝑐0𝑥𝛽+𝑐1𝑥𝛽+1++𝑐𝑛𝛽𝑥𝑛,(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,,𝑐𝑛𝛽 are all of solutions of the following system for 𝑖=0,𝑖=1,,𝑖=𝑛𝛽, respectively, 𝑅𝑡𝑣𝑉1,(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,,𝑐𝑛𝛽. It is trivial that 𝑐𝑛𝛽=1 and this completes the proof.

Algorithm 4.4 (For computation 𝑐𝑖). One has the following.Step 1. Solve 𝛽=min𝑛𝑖=1𝑣𝑖,𝑅𝑡𝑉1𝑚,𝑣𝑖{0,1},where𝑖=1,2,,𝑛,(4.4) and obtain 𝛽.Step 2. For 𝑖=0 to 𝑛𝛽1, compute all of solutions: 𝑅𝑡𝑉1𝑚,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,𝑒𝑗𝐸 with common endpoint 𝑣𝑖 at most 𝑒𝑖 or 𝑒𝑗 belongs to 𝑀. Therefore we have the following.

Theorem 5.1. One has 𝛼=max𝑚𝑖=1𝑒𝑖subjectto𝑅𝐸1𝑛,𝑒𝑗{0,1},where𝑗=1,2,,𝑚.(5.1)

Definition 5.2. A matching polynomial is as follows: 𝑀(𝑥)=𝑑0𝑥𝛼+𝑑1𝑥𝛼1++𝑑𝛼1𝑥,(5.2) where 𝛼 is the same as in (5.1) and 𝑑𝑖’s are the number of matching sets with 𝛼𝑖 elements.

Theorem 5.3. The coefficients 𝑑0,𝑑1,,𝑑𝛼1 are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,,𝑖=𝛼1, 𝑅𝐸1𝑛,𝑒1+𝑒2++𝑒𝑚=𝛼𝑒𝑖,𝑗{0,1},where𝑗=1,2,,𝑚.(5.3)

Algorithm 5.4 (For computation 𝑑𝑖). One has the following.Step 1. Solve 𝛼=max𝑚𝑖=1𝑒𝑖,𝑅𝐸1𝑛,𝑒𝑗{0,1},where𝑗=1,2,,𝑚,(5.4) and obtain 𝛼.Step 2. For 𝑖=0 to 𝛼2, compute all of solutions: 𝑅𝐸1𝑛,𝑒1+𝑒2++𝑒𝑚=𝛼𝑒𝑖,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,,𝛼2, of course 𝑑𝛼1=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=min𝑛𝑖=1𝑣𝑖,subjectto𝐴+𝐼𝑛𝑉1𝑛,𝑣𝑖{0,1},where𝑖=1,2,,𝑛.(6.1)

Definition 6.2. A dominating polynomial is as follows: 𝐷(𝑥)=𝑓0𝑥𝛾+𝑓1𝑥𝛾+1++𝑓𝑛𝛾𝑥𝑛,(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,,𝑓𝑛𝛾 are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,,𝑖=𝑛𝛾, 𝐴+𝐼𝑛𝑉1𝑛𝑣,()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,,𝑓𝑛𝛾. It is trivial that 𝑓𝑛𝛾=1 and this completes the proof.

Algorithm 6.4 (For computation 𝑓𝑖). One has the following.Step 1. Solve 𝛾=min𝑛𝑖=1𝑣𝑖,𝐴+𝐼𝑛𝑉1𝑛,𝑣𝑖{0,1},where𝑖=1,2,,𝑛,(6.3) and obtain 𝛾.Step 2. For 𝑖=0 to 𝑛𝛾1, compute all of solutions: 𝐴+𝐼𝑛𝑉1𝑛,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 𝛾=min𝑚𝑖=1𝑒𝑖subjectto𝐵+𝐼𝑚𝐸1𝑚,𝑒𝑖{0,1},where𝑖=1,2,,𝑚.(7.1)

Definition 7.2. An edge dominating polynomial is a polynomial such as 𝑊(𝑥)=𝑓0𝑥𝛾+𝑓1𝑥𝛾+1++𝑓𝑚𝛾𝑥𝑚,(7.2) where 𝛾 is the same as in (7.1) and 𝑓𝑖’s are the number of edge dominating sets with 𝛾+𝑖 elements.

Theorem 7.3. The coefficients 𝑓0,𝑓1,,𝑓𝑚𝛾 are all of solutions of the following system, respectively, 𝑖=0,𝑖=1,,𝑖=𝑚𝛾, 𝐵+𝐼𝑚𝐸1𝑚,𝑒1+𝑒2++𝑒𝑚=𝛾𝑒+𝑖,𝑗{0,1},where𝑗=1,2,,𝑚.(7.3)

Algorithm 7.4 (For computation 𝑓𝑖). One has the following.Step 1. Solve 𝛾=min𝑚𝑖=1𝑒𝑖,𝐵+𝐼𝑚𝐸1𝑚,𝑒𝑖{0,1},where𝑖=1,2,,𝑚,(7.4) and obtain 𝛾.Step 2. For 𝑖=0 to 𝑚𝛾1, compute all of solutions: 𝐵+𝐼𝑚𝐸1𝑚,𝑒1+𝑒2++𝑒𝑚=𝛾𝑒+𝑖,𝑗{0,1},where𝑗=1,2,,𝑚.(7.5)Step 3. Set 𝑓𝑖 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.
  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.