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UDC 519.54

MATHEMATICS

G. V. EPIFANOV

REDUCTION OF A PLANE GRAPH TO AN EDGE BY STAR–TRIANGLE TRANSFORMATIONS

(Presented by Academician L. V. Kantorovich, 2 VI 1965)

Let us consider the following transformations of undirected multigraphs*.
(I) Replacement of two edges joining one and the same pair of vertices by a single edge. (II) Replacement of a chain of two edges joining a pair of vertices by a single edge joining these same vertices. (III) Replacement of three edges forming a triangle with vertices $a, b, c$ by a star: three edges $[a,d]$, $[b,d]$, $[c,d]$, with a new vertex $d$. (IV) The transformation inverse to (III).

Eikers $(^2)$, and then Lehman $(^3)$, conjectured that every plane multigraph with two poles (distinguished vertices), satisfying certain restrictions natural for transportation networks, can be reduced to the single edge joining these poles by means of a finite number of transformations of types (I)—(IV) preserving the poles. In the present note we set forth the scheme of proof of the following lemma, from which the validity of this conjecture follows.

Main lemma. Let $G$ be a finite connected plane undirected multigraph possessing the following properties: $G$ has no more than two poles and at least one vertex that is not a pole; $G$ has no loops and no vertices of degree 1 distinct from the poles. Then there exists a finite (possibly empty) sequence of transformations of types (III), (IV), preserving the poles and the property of multigraphs of being plane, which transforms $G$ into a plane multigraph $G'$ such that it possesses all the listed properties of the multigraph $G$ and at least one of the transformations (I), (II), preserving the poles, is applicable to it.

The following is easily obtained from the main lemma.

Theorem. Let $G$ be a plane undirected multigraph with two poles. In order that there exist a finite (possibly empty) sequence of transformations (I)—(IV), preserving the poles, reducing $G$ to the single edge joining these poles, it is sufficient that the multigraph $G$ be finite, have no isolated vertices, and that through each of its edges there pass an elementary chain, joining the poles.

From this follows the validity of the conjecture indicated above.

The proof of the main lemma (items 2–12) will be carried out by contradiction. Let $\Gamma$ be the family of all multigraphs satisfying the conditions of the theorem, for which nevertheless its conclusion is false and which moreover have the least possible number of edges. Suppose that $\Pi$ is nonempty. Then all elements of $\Gamma$ are graphs in which every vertex distinct from a pole has degree $>2$; transformations (III), (IV) do not lead out of $\Gamma$.
Considering an arbitrary graph $G \in \Gamma$, given by some realization on the sphere $\Sigma$, we introduce the notion of a completely admissible set—

* In terminology we follow $(^1)$.
** That is, a chain without coincident vertices.

(with respect to the order of the transformations (III), (IV)), and also a number of auxiliary concepts.

Let $\mu=(v_1,v_2,\ldots,v_p)$, $v_i=[b_{i-1},b_i]$, $1\le i\le p$, be an arbitrary (oriented) chain in the graph $G$. The angle from $v_i$ to $v_{i+1}$, $1\le i<s$, counted counterclockwise (respectively clockwise), will be called the right (respectively left) angle of the chain $\mu$ at the vertex $b_i$; the vertex $b_i$, $1\le i<s$, will be called a right (left) vertex of the chain $\mu$, if inside the right (left) angle of the chain $\mu$ at the vertex $b_i$ there are no edges of the graph $G$ incident with this vertex. Let $C=(u_1,u_2,\ldots,u_s)$, $u_i=[a_{i-1},a_i]$, $1\le i\le s$, be such a simple cycle* in the graph $G$ that, if it has coincident vertices, then at them $C$ does not cross itself, but only touches itself; suppose, moreover, that every such “multiple” vertex is either a right or a left vertex of the cycle $C$; then the cycle $C$ will be called a regular cycle. Obviously, a regular cycle $C$, considered as a subset of the sphere $\Sigma$, determines two closed subsets $T_1$ and $T_2$ of it with common boundary $C$, which have the following property: if a sufficiently small neighborhood of any interior point of an arbitrary edge $u_i$, $1\le i\le s$, of the cycle $C$ is taken, then all points of this neighborhood lying to the left of the edge $u_i$ (in the direction of traversal of the cycle $C$) are contained in the set $T_1$, and all points lying to the right are contained in the set $T_2$. Such closed sets $T_1$ and $T_2$ will be called, respectively, the left and right sets determined by the cycle $C$.

Let $T$ be either the left or the right set determined by the cycle $C$. A vertex of the cycle $C$ which is not a pole will be called principal if every edge of the graph $G$ incident with this vertex is contained in $T$. Suppose a certain partition is given of all non-principal vertices of $C$ into two disjoint classes: special vertices and inessential vertices; here the special vertices are chosen from among those vertices for which there exists exactly one incident edge of the graph $G$ lying outside $T$. An edge of the cycle $C$ will be called special (for the given partition) if both its ends are inessential vertices. The peculiarities of the cycle $C$ will mean both its special vertices and its special edges. If the number of peculiarities of the cycle $C$ under the given partition is $<3$, then such a partition will be called admissible. If, moreover, the poles of the graph $G$ are neither inside $T$ nor among the principal or special vertices of $C$, then such a partition will be called completely admissible. The left or right set $T$ determined by the cycle $C$ will be called an admissible (completely admissible) set if for the cycle $C$ there exists an admissible (completely admissible) partition.

We shall show that every graph $G\in\Gamma$ can be realized on the sphere $\Sigma$ in such a way that there is a regular cycle $C$ of the graph $G$ determining a completely admissible set. First we introduce one auxiliary concept. Let $\mu=(v_1,v_2,\ldots,v_p)$, $v_i=[b_{i-1},b_i]$, $1\le i\le p$, be such a chain of the graph $G$, given by some realization on the sphere $\Sigma$, that all its vertices except the initial and terminal ones are, alternately (in the order of occurrence in $\mu$), right and left vertices. Then the chain $\mu$ will be called a perfect chain.

For the proof it is evidently sufficient to consider the case when $G$ has two poles $P_1$ and $P_2$. Suppose that at least one of the poles, for example $P_1$, has degree $>1$. We then construct such a perfect chain
\[ \mu_1=(v_1,v_2,\ldots,v_p),\quad v_i=[b_{i-1},b_i],\quad 1\le i\le p, \]
that: a) $b_0=P_1$; b) $(B_p\equiv P_1)\vee(B_p\equiv P_2)\vee(\exists i_0(1\le i_0<p-1\ \&\ v_p\equiv v_{i_0}))$; c) the chain $\mu_1$ is minimal (with respect to inclusion) among all perfect chains possessing properties a) and b). If $b_p\equiv P_1$, but $v_p\not\equiv v_1$, then $\mu_1$ is a regular cycle and determines two admissible sets, of which one (not containing $P_2$) is completely admissible. In the contrary case we construct

* That is, a cycle without coincident edges.

such a perfect chain $\mu_2=(w_1,w_2,\ldots,w_q)$, $w_j=[c_{j-1},c_j]$, $1\le j\le q$, that: a) $c_0\equiv P_1$; b) $w_1\ne v_1$; c) $(c_q\equiv P_1)\vee(c_q\equiv P_2)\vee(\exists j_0(1\le j_0<q-1\ \&\ w_q\equiv w_{j_0}))\vee(\exists i_0(1\le i_0\le p\ \&\ w_q\equiv v_{i_0}))$; d) if $p>1$ and $q>1$, then the vertices $b_1$ and $c_1$ are simultaneously either right or left; e) the chain $\mu_2$ is minimal (with respect to inclusion) among all perfect chains possessing properties a)—d). It is not difficult to show that at least one of the maximal regular cycles contained either in the chain $\mu_1$, or in the chain $\mu_2$, or in the union of the chains $\mu_1$ and $\mu_2^{-1}=(w_q,w_{q-1},\ldots,w_1)$, determines a completely admissible set.

If both $P_1$ and $P_2$ have degree 1, then for the graph $G_1$, obtained from $G$ by deleting $P_1$ and the edge $[P_1,P'_1]$ (where $P'_1$ is the vertex adjacent to $P_1$), we find a completely admissible set $T$, regarding $P'_1$ and $P_2$ as poles of $G_1$; it remains to return $P_1$ and $[P_1,P'_1]$ in such a way that they lie outside $T$.

We shall call the weight of a completely admissible set $T$ the ordered pair $(k,l)$, where $k$ is the number of edges of the graph $G$ contained in $T$ (including the boundary of $T$), and $l$ is the number of edges lying on the boundary of $T$. We shall assume that
\[ (k,l)<(k',l')\Longleftrightarrow (k<k'\vee(k=k'\ \&\ l<l')). \]
For every graph in $\Gamma$ consider the set of all its possible realizations on the sphere $\Sigma$ and choose a graph $G_0\in\Gamma$ such that in some realization of it there exists a completely admissible set $T_0$ with the least possible weight (among all realizations of all graphs in $\Gamma$).
Assuming that on the cycle $C_0$, which determines the thus chosen $T_0$, the corresponding completely admissible partition is specified, we indicate some properties of $C_0$ and $T_0$. a) Every principal vertex of the cycle $C_0$ has degree $\ge3$. b) No special vertex of $C_0$ having degree 3 can be adjacent to two inessential vertices of $C_0$. c) In the graph $G_0$ there is no such “empty” triangle$^*$ that one of its edges lies inside $T_0$, and the other two lie on $C_0$, moreover the vertex common to these last two edges is an inessential vertex of $C_0$. d) If the interior of $T_0$ is homeomorphic to an open disk, then inside $T_0$ there is an edge of the graph $G_0$. e) If the interior of $T_0$ is not homeomorphic to an open disk, then among the cycles contained in $C_0$ there is at least one cycle $C_0^*$ having no more than one special feature of the cycle $C_0$, such that in it the order of immediate succession of edges coincides with the order of immediate succession of these same edges in the cycle $C_0$, except for one pair of immediately following edges of $C_0^*$; moreover the cycle $C_0^*$ contains no more than one special feature of the cycle $C_0$. Let $T_0^*$ be the left or right set determined by the cycle $C_0^*$ and contained in $T_0$. Then inside $T_0^*$ there is an edge of the graph $G_0$.
We outline the proof of the following assertion. Let $\widehat T$ be any such left or right set determined by a regular cycle
\[ \widehat C=(\widehat u_1,\widehat u_2,\ldots,\widehat u_s),\quad \widehat u_i=[\widehat a_{i-1},\widehat a_i],\quad 1\le i\le s \]
(in the graph $G_0$), with some (not necessarily admissible) partition specified on $\widehat C$, such that $\widehat T$, $\widehat C$, and the given partition satisfy the following conditions: a) the interior of $\widehat T$ is homeomorphic to an open disk; b) there is an $r$, $1\le r\le s$, such that the chain
\[ \widehat\mu=(\widehat u_1,\widehat u_2,\ldots,\widehat u_r) \]
has the following properties: all vertices of the chain $\widehat\mu$ are inessential; for the vertices $\widehat a_0$ and $\widehat a_r$, and only for these vertices of the chain $\widehat\mu$, there exist edges of the graph $G_0$ incident with these vertices and lying inside $\widehat T$; c) $\widehat C$ has $\le1$ special vertices; d) among the edges $\widehat u_i$, $r<i\le s$, there are $\le1$ special edges of the cycle $\widehat C$; if such a special edge exists, then $\widehat C$ has no special vertices; e) all vertices and edges of the cycle $\widehat C$ not lying on the chain $\widehat\mu$ have properties a)—c) indicated in item 6 for the vertices and edges of the cycle $C_0$; f) inside $\widehat T$ there are no poles of the graph $G_0$. Then there exists a completely admissible set contained in $\widehat T$ and not coinciding with $\widehat T$.

$^*$ That is, one of the regions into which this triangle divides the sphere $\Sigma$ does not contain edges of the graph $G_0$.

The proof of the assertion of item 7 is carried out by contradiction (items 8—11). Suppose that there exist sets satisfying all the conditions of item 7 for which, nevertheless, the conclusion of item 7 is false. Choose from among them such a set $\hat T_0$ that contains the minimal number of edges of the graph $G_0$. For definiteness one may assume that $\hat T_0$ is a left set, determined by the cycle $\hat C_0=(u_1,u_2,\ldots,u_s)$, $u_i=[a_{i-1},a_i]$, $1\leq i\leq s$.
Construct a chain $\mu^*=(v_1,v_2,\ldots,v_p)$, $v_j=[b_{j-1},b_j]$, $1\leq j\leq p$, satisfying the following conditions: a) $b_0\equiv a_0$, and this vertex is the left vertex of the chain $(u_s,v_1)$; b) all edges $v_j$, $1\leq j\leq p$, lie inside $\hat T_0$; c) if, for $1<j<p$, the vertex $b_j$ lies inside $\hat T_0$, then it is a left vertex of the chain $\mu^*$; d) if, for $1<j<p$, the vertex $b_j$ lies on $\hat C_0$, then it is a principal vertex of the cycle $\hat C_0$; moreover, there exist exactly two edges of the graph $G_0$ incident with $b_j$ and lying inside the left angle of the chain $\mu^*$ at the vertex $b_j$; e) for $1\leq j<p$ there do not exist vertices $b_{j-1}$ and $b_j$ both lying on $\hat C_0$; f) the chain $\mu^*$ is maximal among all chains satisfying conditions a)—e).
Let $b_{j_1}$, $b_{j_2}$, $0\leq j_1<j_2\leq p$, be two “neighboring” vertices of the chain $\mu^*$ lying on $\hat C_0$; $b_{j_1}\equiv a_{i_2}$, $b_{j_2}\equiv a_{i_1}$, $0\leq i_1<i_2\leq s$. Using the minimality of $\hat T_0$, one can show that inside the right set determined by the cycle $C_{i_1}=(v_{j_2},v_{j_2-1},\ldots,v_{j_1+1},u_{i_2},u_{i_2-1},\ldots,u_{i_1+1})$ there are no edges of the graph $G_0$.
Using the minimality of $\hat T_0$ and item 10, one can show that in the left set $T'$, determined by the cycle $C'=(u_1,u_2,\ldots,u_{i_0},v_p,v_{p-1},\ldots,v_1)$ $(a_{i_0}\equiv b_p)$, there is contained a fully admissible set, which contradicts the assumption about $\hat T_0$.
If the interior $T_0$ (item 5) is homeomorphic to an open disk, then, using item 6, it is easy to show that $T_0$ satisfies all the conditions of item 7 for the set $\hat T$; if it is not homeomorphic, these conditions are satisfied by the set $T_0^*$ (item 6). But then the conclusion of item 7, respectively for $T_0$ or $T_0^*$, contradicts the assumption of the minimality of the weight of $T_0$.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
30 V 1965

REFERENCES

$^{1}$ C. Berge, The Theory of Graphs and Its Applications, 1962.
$^{2}$ S. B. Akers, Operations Res., 8, 3, 311 (1960).
$^{3}$ A. Lehman, J. Soc. Ind. and Appl. Math., 11, 3, 773 (1963).

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[v1] 1966-01-01

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REDUCTION OF A PLANE GRAPH TO AN EDGE BY STAR–TRIANGLE TRANSFORMATIONS

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UDC 519.54