MATHEMATICS

V. G. VIZING

EQUIVALENT FLOWS IN A TRANSPORT NETWORK

(Presented by Academician S. L. Sobolev, 5 II 1962)

A. Let there be given a finite transport network \(G(X,U)\) with input \(x_0\), output \(z\), and capacities \(c(u)\ge 0\), where \(u\in U\) (see (1), Ch. VIII). Two flows \(\varphi\) and \(\varphi'\) in the network \(G\) are called equivalent if \(\varphi_z=\varphi'_z\). The distance between the flows \(\varphi\) and \(\varphi'\) will be the quantity

\[ \rho(\varphi,\varphi')=\sum_{u\in U}|\varphi(u)-\varphi'(u)|. \]

A \((\varphi,\varphi')\)-cycle will mean an elementary cycle composed of arcs \(u\in U\) such that on arcs of one direction \(\varphi(u)<\varphi'(u)\), and on arcs of the opposite direction \(\varphi(u)>\varphi'(u)\). A \(\varphi\)-cycle will mean an elementary cycle composed of arcs \(u\in U\) such that on arcs of one direction \(\varphi(u)<c(u)\), and on arcs of the opposite direction \(\varphi(u)>0\).

Theorem 1*. Let \(\varphi\) be a flow in the network \(G(X,U)\). For the existence of an equivalent flow \(\varphi'\) such that \(\rho(\varphi,\varphi')>0\), it is necessary and sufficient that a \(\varphi\)-cycle exist.

Sufficiency. Suppose that a \(\varphi\)-cycle exists. Put: \(\varphi'(u)=\varphi(u)+1\) on those arcs of the cycle on which \(\varphi(u)<c(u)\); \(\varphi'(u)=\varphi(u)-1\) on those arcs of the cycle on which \(\varphi(u)>0\); \(\varphi'(u)=\varphi(u)\) on the remaining arcs of the transport network \(G\).

We have \(\varphi_z=\varphi'_z\) and \(\rho(\varphi,\varphi')>0\).

Necessity. Suppose that there exist two equivalent flows \(\varphi\) and \(\varphi'\) in the network \(G\), with \(\rho(\varphi,\varphi')>0\). Since a \((\varphi,\varphi')\)-cycle is obviously a \(\varphi\)-cycle, it is enough to prove the existence of a \((\varphi,\varphi')\)-cycle.

If on \(u_1\in U_z^{-}\) (or \(u_1\in U_{x_0}^{+}\)) \(\varphi(u_1)>\varphi'(u_1)\), then, owing to the equivalence of the flows \(\varphi\) and \(\varphi'\), there exists an arc \(u_2\in U_z^{-}\) \((u_2\in U_{x_0}^{+})\) on which \(\varphi(u_2)<\varphi'(u_2)\). If, however, for \(x\ne x_0\), \(x\ne z\), there exists an arc \(u\in U_x^{-}\) on which \(\varphi(u)>\varphi'(u)\) \((\varphi(u)<\varphi'(u))\), then, since \(\varphi\) and \(\varphi'\) are flows and the transport network has no loops, there exists a vertex \(y\), distinct from \(x\), such that:

\[ \text{either }\quad \varphi(x,y)>\varphi'(x,y)\quad \bigl(\varphi(x,y)<\varphi'(x,y)\bigr); \]

\[ \text{or }\quad \varphi(y,x)<\varphi'(y,x)\quad \bigl(\varphi(y,x)>\varphi'(y,x)\bigr). \]

Thus there exists an infinite \((\varphi,\varphi')\)-chain; and since the transport network is finite, the chain closes, and the first closure gives us the required \((\varphi,\varphi')\)-cycle.

Theorem 1 is proved.

B. From the proof of Theorem 1 it follows that if \(\varphi'_z=\varphi_z\) and \(\rho(\varphi,\varphi')>0\), then the flow \(\varphi'\) can be obtained from \(\varphi\) in the following way.

Find a \((\varphi,\varphi')\)-cycle. Put:

\[ \begin{aligned} \varphi''(u)&=\varphi(u)+1 &&\text{on those arcs of the cycle on which } \varphi(u)<\varphi'(u);\\ \varphi''(u)&=\varphi(u)-1 &&\text{on those arcs of the cycle on which } \varphi(u)>\varphi'(u);\\ \varphi''(u)&=\varphi(u) &&\text{on the remaining arcs of the transport network.} \end{aligned} \]

* Note added in proof. After the paper had been submitted for publication, it became known to the author that, in a somewhat more general form, a theorem of analogous content was proved in the work of Hwang Tua (2).

We have: \(\varphi''_z=\varphi_z-\varphi'_z\) and \(\rho(\varphi'',\varphi')<\rho(\varphi,\varphi')\).
If \(\rho(\varphi'',\varphi')>0\), then we find a \((\varphi'',\varphi')\)-cycle, construct a flow \(\varphi'''\), etc. Since \(\rho(\varphi,\varphi')\) is a nonnegative integer, after a finite number of such steps we obtain a flow \(\widetilde{\varphi}\) such that \(\rho(\widetilde{\varphi},\varphi')=0\).

B. Here we shall generalize to \(p\)-graphs Theorem 2 from Chapter IX \((^1)\).

Let there be a \(p\)-graph \(G=(X,U)\) with vertices \(\{x_i\}\), \(i=1,2,\ldots,n\), and semidegrees \((r_i,s_i)\), \(i=1,2,\ldots,n\) (\(r_i\) is the out-semidegree, \(s_i\) the in-semidegree of the vertex \(x_i\)).

Denote by \(\alpha_G(x,y)\) the number of arcs in \(U\) going from \(x\) to \(y\). Let \(x,y,x',y'\) be vertices of \(G\), with \(x\ne x'\), \(y\ne y'\); \(\alpha_G(x,y)>0\); \(\alpha_G(x',y')>0\); \(\alpha_G(x,y')<p\); \(\alpha_G(x',y)<p\).

We shall say that a \(p\)-graph \(G'=(X,V)\) is obtained from \(G=(X,U)\) by one transfer if it differs from \(G\) only in that

\[ \alpha_{G'}(x,y)=\alpha_G(x,y)-1;\qquad \alpha_{G'}(x',y')=\alpha_G(x',y')-1; \]

\[ \alpha_{G'}(x,y')=\alpha_G(x,y')+1,\qquad \alpha_{G'}(x',y)=\alpha_G(x',y)+1. \]

It is clear that a transfer carries a \(p\)-graph into a \(p\)-graph with the same semidegrees.

Theorem 2. Let \(G=(X,U)\) and \(H=(X,V)\) be two \(p\)-graphs with respectively equal semidegrees. Then \(H\) can be obtained from \(G\) by a finite number of transfers.

Construct a transportation network \(T\) with input \(x_0\), output \(z\), and vertices
\[ x_1,x_2,\ldots,x_n,\ \bar{x}_1,\bar{x}_2,\ldots,\bar{x}_n. \]
Join the vertices \(x_i\) and \(\bar{x}_j\) by an arc of capacity \(c(x_i,\bar{x}_j)=p\), the vertices \(x_0\) and \(x_i\) by an arc of capacity \(c(x_0,x_i)=r_i\), and the vertices \(\bar{x}_j\) and \(z\) by an arc of capacity \(c(\bar{x}_j,z)=s_j\). Then the \(p\)-graph \(G\) determines a flow \(\varphi\) in the network \(T\), saturating the input and output arcs, with \(\varphi(x_i,\bar{x}_j)=\alpha_G(x_i,x_j)\); the \(p\)-graph \(H\) correspondingly determines a flow \(\varphi'\), of the same value as \(\varphi\). The theorem will be proved if we show that, by a finite number of transformations of the flow equivalent to transfers in the \(p\)-graph, one can obtain from \(\varphi\) a flow \(\widetilde{\varphi}\) of the same value as \(\varphi\) and such that \(\rho(\widetilde{\varphi},\varphi')<\rho(\varphi,\varphi')\).

Obviously, all \((\varphi,\varphi')\)-cycles will consist of arcs of the form \((x_i,\bar{x}_j)\). Consider an alternating \((\varphi,\varphi')\)-cycle, which, for simplicity, we denote by

\[ \mu=[(x_1,y_1),(x_2,y_1),(x_2,y_2),(x_3,y_2),\ldots,(x_m,y_m),(x_1,y_m)]. \]

The vertices \(x_i\) and \(x_{i+1}\) for \(i<m\), and also \(x_m\) and \(x_1\), will be called adjacent. We shall assume that
\[ \varphi(x_i,y_i)>\varphi'(x_i,y_i),\quad i=1,2,\ldots,m; \]
\[ \varphi(x_{i+1},y_i)<\varphi'(x_{i+1},y_i),\quad i=1,2,\ldots,m-1;\qquad \varphi(x_1,y_m)<\varphi'(x_1,y_m). \]

For each vertex \(x_j\), denote by \(\Gamma_{\varphi,\varphi'}(x_j)\) the set of vertices among \(\{y_i\}\), \(i=1,2,\ldots,m\), which are the ends of arcs going out of \(x_j\), on which either \(\varphi>\varphi'\) or \(\varphi=p\). Obviously, \(y_j\in\Gamma_{\varphi,\varphi'}(x_j)\).

There exist two adjacent vertices \(x\) and \(x'\) such that
\[ \Gamma_{\varphi,\varphi'}(x')\not\subset \Gamma_{\varphi,\varphi'}(x) \quad\text{and}\quad \Gamma_{\varphi,\varphi'}(x)\not\subset \Gamma_{\varphi,\varphi'}(x'). \]
Indeed, otherwise, since \(y_i\notin\Gamma_{\varphi,\varphi'}(x_{i+1})\) for \(i<m\) and \(y_m\notin\Gamma_{\varphi,\varphi'}(x_1)\), we would obtain
\[ \Gamma_{\varphi,\varphi'}(x_1)\supset\supset \Gamma_{\varphi,\varphi'}(x_2)\supset\supset\cdots\supset\supset \Gamma_{\varphi,\varphi'}(x_m)\supset\supset \Gamma_{\varphi,\varphi'}(x_1), \]
which contradicts the notion of strict inclusion.

Without loss of generality, suppose that
\[ \Gamma_{\varphi,\varphi'}(x_1)\not\subset \Gamma_{\varphi,\varphi'}(x_2) \quad\text{and}\quad \Gamma_{\varphi,\varphi'}(x_2)\not\subset \Gamma_{\varphi,\varphi'}(x_1). \]
We have: \(y_1\in\Gamma_{\varphi,\varphi'}(x_1)\); \(y_1\notin\Gamma_{\varphi,\varphi'}(x_2)\), and for some \(k\ge 2\),
\[ y_k\in\Gamma_{\varphi,\varphi'}(x_2),\qquad y_k\notin\Gamma_{\varphi,\varphi'}(x_1). \]
Put:

\[ \varphi_1(x_1,y_1)=\varphi(x_1,y_1)-1,\qquad \varphi_1(x_2,y_k)=\varphi(x_2,y_k)-1; \]

\[ \varphi_1(x_1,y_k)=\varphi(x_1,y_k)+1,\qquad \varphi_1(x_2,y_1)=\varphi(x_2,y_1)+1, \]

\[ \varphi_1(u)=\varphi(u)\quad \text{on the remaining arcs of the network } T. \]

We obtain a flow \(\varphi_1\) of the same value as \(\varphi\). Such a transformation of the flow is equivalent to a transfer in the \(p\)-graph. Obviously,
\[ \rho(\varphi_1,\varphi')\le \rho(\varphi,\varphi'). \]

If \(\rho(\varphi_1,\varphi') < \rho(\varphi,\varphi')\), then the required assertion is proved. If, however, \(\rho(\varphi_1,\varphi') = \rho(\varphi,\varphi')\), then, as before, we consider the \((\varphi_1,\varphi')\)-cycle

\[ \mu_1=[(x_1,y_k),(x_{k+1},y_k),(x_{k+1},y_{k+1}),\ldots,(x_m,y_m),(x_1,y_m)], \]

which already consists of a smaller number of arcs, and in the same way perform a transfer. After a certain number of steps we either reduce \(\rho(\varphi,\varphi')\), or obtain a flow \(\varphi''\) such that \(\rho(\varphi'',\varphi')=\rho(\varphi,\varphi')\) and the \((\varphi'',\varphi')\)-cycle consists of four arcs. But then by the next transfer we transform the flow \(\varphi''\) into a flow \(\widetilde{\varphi}\) such that

\[ \rho(\widetilde{\varphi},\varphi')=\rho(\varphi'',\varphi')-4 =\rho(\varphi,\varphi')-4<\rho(\varphi,\varphi'). \]

Theorem 2 is proved.

G. Theorem 3, given below, solves the problem posed in (1) (Appendix IV) concerning the maximum matchings of a simple graph. Let us recall several definitions.

A matching of a graph \((X,Y,\Gamma)\) is a set \(W\) of its arcs in which no two arcs are adjacent. The arcs of the matching \(W\) will be called strong, and the remaining arcs of the graph \((X,Y,\Gamma)\) weak; the endpoints of strong arcs will be called saturated vertices.

Let \(W\) be a maximum matching, and let \(\mu\) be an elementary cycle of even length formed by alternating strong and weak edges. Then a new maximum matching may be defined as follows:
\[ W'=(W\setminus\mu)\cup(\mu\setminus W). \]
We shall call such an operation a cyclic transfer.

Theorem 3. Let \(W\) and \(W'\) be two maximum matchings of a simple graph \((X,Y,\Gamma)\). In order that \(W'\) can be obtained from \(W\) by a finite number of cyclic transfers, it is necessary and sufficient that \(W\) and \(W'\) have the same saturated vertices.

Necessity follows from the fact that a cyclic transfer neither destroys old saturated vertices nor creates new saturated vertices.

Sufficiency. Consider the transportation network \(G\) defined by the graph \((X,Y,\Gamma)\), with an input \(x_0\) connected to each vertex \(x_i\in X\), and an output \(z\) connected to each vertex \(y_i\in Y\); all arcs of the transportation network have capacity \(c(u)=1\).

If \(\varphi\) is some maximum flow in \(G\), then the arcs of the graph \((X,Y,\Gamma)\) on which \(\varphi=1\) form a maximum matching. Conversely, to every maximum matching there corresponds some maximum flow in \(G\). Now the sufficiency follows immediately from A and B.

Theorem 3 is proved.

I express my gratitude to A. A. Zykov for posing the questions considered in this paper.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
31 I 1962

CITED LITERATURE

1. C. Berge, Theorie des graphes et ses applications, Paris, 1958.
2. Hoàng, Tụy, Toán Lý Hóa, No. 3 (1961) (in Vietnamese).