On Total Unimodularity of Edge–Edge Adjacency Matrices

Abstract

We consider total unimodularity for edge–edge adjacency matrices that represent adjacency relations between pairs of edges in a graph. These matrices appear in integer programming formulations of the minimum maximal matching problem, the edge dominating set problem, and so on.

We investigate the problem of characterizing graphs that have totally unimodular edge–edge adjacency matrices, and give a necessary and sufficient condition for characterization. This condition is the first characterization for total unimodularity of edge–edge adjacency matrices.

Appendix

In this appendix, we give alternative proofs of Lemmas 1–3.

Given a matrix A, if every square non-singular submatrix of A is an integer matrix with determinant 1 or −1, we say that A is totally unimodular. From this definition, it is clear that if a submatrix of A is not totally unimodular, A is not totally unimodular. In the alternative proofs, we show that each EE matrix of C _t for t≥4, T _S , and G _TC has a square non-singular submatrix with determinant other than −1 and 1.

We use the following lemma to prove Lemma 1.

Lemma 7

([9], Lemma 9.2.4 on p. 190)

The following matrix is totally unimodular if and only if its entries sum to $0\ (\operatorname{mod} 4)$.

$$ \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \pm 1 &\pm 1 & & & \\&\pm 1 &\pm 1 & \huge{0} & \\& &\ddots &\ddots & \\&\huge{0} & &\pm 1 &\pm 1 \\\pm 1 & & & &\pm 1 \end{array} \right ) $$

(5)

In the alternative proof of Lemma 1, we find a submatrix which has the same form as the matrix (5) and is not totally unimodular.

Lemma 1

The EE matrix M(C _t ) for t≥4 is not totally unimodular.

Proof

Let e ₁ be any edge in C _t . In counter clockwise direction from e ₁, we assign labels e ₂ to e _t to edges other than e ₁ (Fig. 12). Then, M(C _t ) is represented by

Fig. 12

C _k

Let k be a positive integer. We classify t≥4 into four cases t=4k,4k+1,4k+2,4k+3.

First, we consider M(C _4k+2). Let $E_{even}^{4k+2}$ be {e _i |i=2j,1≤j≤2k+1}, and $E_{odd}^{4k+2}$ be $E - E_{even}^{4k+2}$. A submatrix of M(C _4k+2) formed by rows $E_{even}^{4k+2}$ and columns $E_{odd}^{4k+2}$ is of the form of the matrix (5), and the sum of the entries of the matrix is 4k+2. Since 4k+2 is equal to $2\ (\operatorname{mod} 4)$, by using Lemma 7, we obtain that the matrix is not totally unimodular. Thus, M(C _4k+2) is not totally unimodular. For instance, the following matrix represents the case k=2, and it can check that the total of the entries of the matrix is equal to 10 (=4∗2+2).

Second, we consider M(C _4k ). The determinant of M(C ₄) is −3, and M(C ₄) is not totally unimodular. We consider M(C _4k ) for k≥2. Let $E_{even}^{4k}$ be {e ₂,e ₃,e ₅}∪{e _i |i=2j,3≤j≤2k} and $E_{odd}^{4k}$ be $E - (E_{even}^{4k} - \{ e_{2}, e_{5} \} )$. A submatrix of M(C _4k ) formed by rows $E_{even}^{4k}$ and columns $E_{odd}^{4k}$ has the form of the matrix (5), and the sum of the entries of the submatrix is 4k+2. Thus, M(C _4k ) is not totally unimodular. For instance, the following matrix represents the case k=2, and it can check that the total of the entries of the matrix is equal to 10.

Third, we consider M(C _4k+1). Let $E_{even}^{4k+1}$ be {e _4k+1}∪{e _i |i=2j,1≤j≤2k} and $E_{odd}^{4k+1}$ be $E - (E_{even}^{4k+1} - \{e_{4k+1}\})$. A submatrix formed by rows $E_{even}^{4k+1}$ and columns $E_{odd}^{4k+1}$ is of the form of the matrix (5), and the sum of the entries of the submatrix is 4k+2. Thus, M(C _4k+1) is not totally unimodular. For instance, the following matrix represents the case k=2, and it can check that the total of the entries of the matrix is equal to 10.

Finally, we consider M(C _4k+3). Let $E_{even}^{4k+3}$ be {e ₂,e ₃,e ₅,e _4k+3}∪{e _i |i=2j,3≤j≤2k+1} and $E_{odd}^{4k+3}$ be $E - (E_{even}^{4k+3} - \{ e_{2}, e_{5}, e_{4k+3} \})$. A submatrix formed by rows $E_{even}^{4k+3}$ and columns $E_{odd}^{4k+3}$ has the form of the matrix (5), and the sum of the entries of the submatrix is 4(k+1)+2. Thus, M(C _4k+3) is not totally unimodular. For instance, the following matrix represents the case k=2, and it can check that the total of the entries of the matrix is equal to 14 (=4∗(2+1)+2).

In earlier discussions, it is clear that M(C _t ) for t≥4 is not totally unimodular. □

Lemma 2

The EE matrix M(T _S ) is not totally unimodular.

Proof

We discuss the EE matrix corresponding to T _S , M(T _S ). Labels e ₁ to e ₆ are assigned to edges in T _S as Fig. 13. Then, M(T _S ) is represented by

The determinant of M(T _S ) is 2. Thus, M(T _S ) is not totally unimodular.

Fig. 13

T _S

 □

Lemma 3

The EE matrix M(G _TC ) is not totally unimodular.

Proof

We discuss the EE matrix corresponding to G _TC , M(G _TC ). Labels e ₁ to e ₆ are assigned to edges in G _TC as Fig. 14. Then, M(G _TC ) is represented by

The determinant of the submatrix formed by rows {e ₄,e ₅,e ₆} and columns {e ₁,e ₂,e ₃} is 2. Consequently, M(G _TC ) is not totally unimodular.

Fig. 14

G _TC

 □