SUBSEMI-EULERIAN GRAPHS

A graph is subeulerian if it is spanned by an eulerian supergraph. Boesch, Suffel and Tindell have characterized the class of subeulerian graphs and determined the minimum number of additional lines required to make a subeulerian graph eulerian.

esting questions concerning related concepts.
The Chinese Postman Problem [1,2] is a celebrated example of such a question.This problem is concerned with the minimum number of repetitions of lines of a graph G which are required if one is to traverse the graph in such a way as to visit each line at least once.Boesch, Suffel and Tindell  [3,4] considered the related question of when a non-eulerian graph can be made eulerian by the addition of lines.
In this paper we consider the problem of when a nonsemi-eulerian graph can be made semi-eulerian by the addition of lines.We characterize "subsemi-eulerian" graphs, i.e., those nonsemi-eulerian graphs which are spanning subgraphs of semi-eulerian graphs.Generalizing the notion of a pairing of points of a graph first introduced by Goodman and Hedetniemi [5] in their study of the Chinese Postman Problem and then used by Boesch, Suffel and Tindell in their study of "subeulerian graphs", we specify the minimum number s+(G) of lines which must be added to a subsemi-eulerian graph to obtain a semi-eulerian spanning supergraph.
We will consider the following problems: i) given a (possibly disconnected) multigraph, when is it possible to obtain a semi-eulerian spanning super multi-graph by the addition of lines and what is the minimum number of required lines?
2) given a (possibly disconnected) graph, when is it possible to obtain a semi- eulerian spanning supergraph by the addition of lines (necessarily from the comple- ment of the graph) and what is the minimum number of required lines?
Although the multigraphs problem is rather easily answered, its solution pro- vides valuable insight into the second problem.The second problem is treated both for connected and disconnected graphs.
An answer to the minimum number of additional lines question is given in both cases.

PREL IMINARIES
In this paper, we make use of standard graph-theoretic notions, terminology, and notation set forth in the book by Harary [6].For the sake of completenes we repeat some of the basic ideas and reintroduce some pertinent notions and results from a previous paper [4].
A graph G (V,X) has a finite point set V and a set X whose elements, called lines, are two-point subsets of V. A multigraph is defined similarly except that more than one line is permitted between two points.A graph V, Y X and the elements of Y join points of U. H s_ G if U V. The degree of a point is a multi-graph is the number of lines incident to it.
A walk W in a graph is an alternating sequence of points and lines v0, Xl, Vl, x2, v 2, Vn_l, Xn, Vn such that x.1 has end points vi_ 1 and vi for i i, ...,n.
In a graph, the notation for a walk is redundant, so we shall shorten it to Vo,Vl,..., v n A graph is connected if, for each pair of points there is a walk joining them.
A maximally connected subgraph of a graph is a component of the graph.A walk W: v 0, x I, v I, Vn_I, x n, Vn is closed if v 0 Vn.
A trail is a walk for which x. # x0 whenever i # j, i.e., no line appears i J more than once.
An eulerian trail is a closed trail which contains all the lines of the graph and an eulerian graph is one which has an eulerian trail.In 173g, Euler proved the first theorem of graph theory: A multigraph is eulerian iff it is connected and each point has even degree.Of those multigraphs which are not eulerian, the ones which can be made eulerian by the addition of lines are called subeulerian.
The minimum number of lines needed to produce a spanning eulerian supermultigraph of the multigraph M is called the eulerian completion number and is denoted by + e (M).A subeulerian graph G is a noneulerian graph which is contained in a spanning eulerian supergraph.The minimum number of lines which must be added to G is de- noted by e+(G).If P0 denotes the number of points of the noneulerian multigraph M having odd degree and k the number of components of M containing only points of even e degree, then  1 Now a path in a multigraph is a walk P: v0,xl,vl,...,Vn_l,Xn,Vn with proper- ty that v. # v. whenever i # j i.e., no repeated points.The points v 0 and v are called the end points of P and we write p {v0'Vn }" A collection p of paths from the graph L with S U P and SP n P' for all pairs of distinct paths Pgp P, P' g P is referred to as a pairing of S on L. If G (V,X) is a graph, then G (V,X) denotes the complement of G, i.e., x X iff x X.We will denote by @(G) the set of all odd degree points of G.A pairing P of @(G) on G will hence- forth be referred to simply as a pairing.Pairings are needed to "complete" con- netted subeulerian graphs to eulerian graphs.
A minimum pairing of @(G on is a pairing P with the property that Pgp is minimum over all pairings of @(G) on (IX(P) denotes the number of lines in P).
The number of lines in such a pairing is denoted by m(@(G), G.
The complete bipartite graph K consists of a point set V partitioned into m,n two parts U and W with IUI m, IVl n and all possible lines joining a point of U to a point of W but no connections internal to U or W.
Complete multipartite graphs are defined similarly; the only difference being that the point set V is partitioned into more than two disjoint parts.THEOREM 2. (SUBEULERIAN CONNECTED GRAPHS) The following are equivalent for a connected graph G: i) G is not subeulerian.
2) G is spanned by K2m+l,2n+I for some value of m.
3) G has evenly many points and G has at least two components with an odd number of points.
If G is subeulerian, then e+(G) m(@(G), ).0If a graph is disconnected, then each pair of its points may be joined in the complement by a path of length no great- er than two.Thus, pairings of odd degree points of a disconnected graph may always employ paths of length no more than two.
The complete graph K consists of a point set with p points and all possible P lines between distinct pairs of points.
The union of two graphs H (VH,XH) and L (VL,) is the graph H u L (VH u VL, u ).THEOREM 3. (SUBEULERIAN DISCONNECTED GRAPHS) With the sole exception of K 1 u n+l' all disconnected graphs are subeulerian.If r denotes the number of paths of length two required in a minimum pairing of the odd degree points of the disconnected graph G, which is not the union of two eulerian components, then # K I u K2n+l and G E 1 u E 2 where E 1 and E 2 are eulerian, then e+(G) 4 if E 1 and E 2 are complete; 3 otherwise.
A multigraph may fail to be eulerian and yet may have a trail which includes all the lines of the multigraph.This is the case when the trail is not closed.
Such a graph is referred to as semi-eulerian and it is known that a .multigraph is semi-eulerian iff it is connected and has exactly two points of odd degree.Of those nonsemi-eulerian multigraphs, the ones which are contained in some spanning semi-eulerian super-multigraph are called subsemi-eulerian.The minimum number of additional lines required by a subsemi-eulerian multigraph M to produce a semi- eulerian super-multigraph is called the semi-eulerian completion number and is + denoted by s (M.The notion of subsemi-eulerian graph and the symbol s+(G)are defined similarly.

SUBSEMI-EULERIAN MULT IGRAPHS
As was stated in Theorem i, every non-eulerian multigraph M is sub-eulerian and e+(M) P0/2 + ke.Thus, by the addition of all but one of the lines required to make M eulerian, i.e., with Po/2 + ke i lines, we may create a semi-eulerian spanning super-multigraph of M. Furthermore, if M is a semi-eulerian spanning super-multigraph of M, only one line need be added to M to obtain a spanning eulerian super-multigraph of M. Thus, it follows that  eulerian multigraph M is also subsemi-eulerian with s+(M) i. 4. CONNECTED SUBSEMI-EULERIAN GRAPHS.
First, we observe that if G is complete then it is certainly impossible to add lines to G from G to make G semi-eulerian.On the other hand, it is clear that any subeulerian graph, connected or not, which is not already semi-eulerian, can be made semi-eulerian by the addition of all but one of the lines of a set which would render the graph eulerian.What if the connected graph G is neither subeulerian not semi-eulerian?Then, by Theorem 2, G is spanned by some K2m+l,2n+I, or equi- valently, G has an even number of points and at least two of the components of G have an odd number of points.Suppose that G has exactly two components with an odd number of points.Then, denoting the components of G with an odd number of points by G 1 and G2, those with an even number of points by G3,...,Gn, and the odd degree points of G lying in G i by O@ for i 1 ...,rwe note that 01 and 02 have odd cardinality while 03,...,0n each have an even number of points.Thus, if P1 is a pairing of all but one point of 01 on G I, P2 is a pairing of a.ll but one point of 02 on G 2 and Pi is a pairing of the set 0.1 on K for i 3,...,nit follows that GU u P is semi-eulerian.It is natural to conjective that, in this case, PgP.
rain m (01 {u}, G I) + rain m(02 {u}, G 2) + m (0i,Gi), (4.1)   u01 ue02 i=3 This is indeed the case but, before we can be sure of this, we must first show that the additional lines needed to make a connected subsemi-eulerian graph a semi- eulerian graph must include a pairing of all but two of the odd degree points of G on G.The reason for concern in this regard is that it is possible for a spanning semi-eulerian supergraph of a graph G to have two odd degree points which were originally even in G.
DEFINITION i. (SEMI-PAIRINGS) If P {PI'''" 'Pn is a collection of paths with lines from G, with u P consisting of all but two points of 0(G), and PgP 8Pi o SP.
for i # j, then P is a semi-pairing of 0(G) on G. J Now, with the aid of this formal notion, we establish LEMMA 5a.(SEMI-EULERIAN SUPERGRAPHS CONTAIN SEMI-PAIRINGS) If is a spanning semi-eulerian supergraph of the subsemi-eulerian (possibly disconnected) non- eulerian graph then -G contains a semi-pairing of @(G) on .
PROOF.Remove all cycles of G G from G and denote the resulting graph by H. Now H is a spanning (possibly disconnected) supergraph of G which has precisely two odd degree points.Furthermore, H-G is a forest so there are line disjoint paths PI"'''Pn with SPi n P.j for each pair of distinct indices i and j and n H-G u P.. To see that this is the case, choose PI to be a longest path in H-G, i=l i P2 to be a longest path in H-(G u P1 ), and so on until all lines of H-G have been accounted for.Now, if P {u v i} and each of the points u i and v have odd de- i i' i gree in G, then P1 'Pn is a semi-pairing of @(G) on G. Suppose there exists an index i such that Uol and v.1 have even degree in G. Then P {Pl,...,Pi_l,Pi+l,...Pn is a pairing of @(G) on G for all the points distinct from u i and v i must be even in H.Of course, removal of any path from P yields a semi-pairing of @(G) on G.
Finally, if there are indices i and j such that u. and u. have even degree in G and v. and v. have odd degree in G, then P' {Pklk # i j} is a semi-pairing of @(G) on G.
1 j Now, in returning to the original problem; we see that, if G has exactly two components with an odd number of points, a semi-pairing of @(G) on G must leave exactly one point in each of the odd components with odd degree.Furthermore, it is also clear that, if the graph G on an even number of points has four or more corn- ponents with an odd number of points, no semi-pairing exists.Finally, if G is sub- eulerian and noneulerian, then a semi-pairing would have to leave two odd degree points in a single component of G. Thus, in this case, s+(G) rain rain (m(@.-{u,v},G i) + E m(@j, Gj)) (4.2) l<i<n u,vc@ 1 i j#i (where @. the collection of odd degree points of G within the component G i of G).
Summarizing the foregoing discussion, we have THEOREM 5. (SUBSEMI-EULERIAN CONNECTED GRAPHS) The following are equivalent for a connected graph G: (i) G is not subsemi-eulerian.
(2) G is complete or G is spanned by a complete multipartite graph having four odd parts.
(3) G is complete or G has an even number of points and G has four or more components with an odd number of points.
If G is subsemi-eulerian, then s+(G) i if G is eulerian or is given by (2) or (I), depending on whether G is subeulerian or not.
Before proceeding to the disconnected case, let us consider the natural ques- tion of whether, for a subeulerian graph, a minimum semi-pairing may be extracted from a minimum pairing by removing a longest path from the minimum pairing?Doubt is cast on the validity of such a conjecture when the possibility of completing a subeulerian graph to a semieulerian nonsubeulerian graph is granted.We illustrate this situation in our first example.and it is easy to see that the paths PI v0,vl,...,Vk+ I and PI v0'vl'''''Vk+l constitute a minimum pairing of the odd degree points of G on G. Thus m(@(G),G) 2(k+l) and it readily follows that m(@(G),G) less the length of a longest path '} is k + i On the other hand, the path P in the minimum pairing P {Pi,Pl Vk+I u,w,vk+ I serves as a minimum semi-pairing of @(G) on G which when added to G yields a non-subeulerian graph.Thus s+(G) 3 < k + i.Furthermore, the difference be- tween the numbers k + i and 3 attains all positive integral values as k ranges over the integers _> 3.
Next suppose P is a minimum semi-pairing of @(G) on G such that Gu u P is PeP subeulerian.Can P be enlarged to a pairing of @(G) on G which contains a path of length long enough to make the size of the pairing larger than m(@(G), G)? EXAMPLE 2. MINIMUM SEMI-PAIRING WHICH CANNOT BE EXTENDED TO MINIMUM PAIRINGS.
A minor modification of the graph given in Example I yields a graph G on 2k + 6 points (k > 3) with complement as shown in Figure 2. All the conclusions reached in Example i, save for one, are valid here.
to G yields a semi-eulerian graph In this case, the addition of P Vk+l,U,W,Vk+ 1 which is subeulerian.The graph G with complement shown in Figure 3 is an example of subsemi-eulerian graph on an odd number of points which contains a minimum semi- pairing that cannot be extended to minimum pairing.In this case, m(@(G),G) 2k+l (k > 4) so that the value of m(@(G),G) less the length of a longest path in a rain- + imum pairing is k.Of cours s (G) 3 so that k > s+(G) whenever k -> 4.

DISCONNECTED SUBSEMI-EULERIAN GRAPHS.
We recall that every disconnected graph with the sole exception of K I u K2n+l (n > 0) is subeulerian (Theorem 3).Because the addition of any line from K 1 to a point of K2n+l yields a semi-eulerian graph, all disconnected graphs are subsemi- eulerian.Furthermore, since graphs are also multigraphs, it follows by Theorem 1 1 + 1 that s+(G) > P0+k I.In fact in some instances s (G) P0 + k i. e e In order to completely determine s+(G), we first verify the following facts: if G is disconnected, then As the final major step in the development, we show that one of the two values 1 P0 + ke i and m(Q(G),G) 2 is always attained.Indeed if P0 0, so that G is the union of two or more eulerian components, G may be made semi-eulerlan by the addition of a path of length k I -P0 + k l, (See Figure (4)).Now, it was stated in the section of preliminaries that, whenever the number of paths of length two of a minimum pairing of G is less than or equal to k e+(G) i +(G) P0 + ke.Hence, in this case, s P0/2 + ke l.On the other hand, if the number of paths of lengths in a minimum pairing exceeds k, then e+(G) m(@(G),).
e Furthermore, if P is a minimum pairing of (G) on G such that H=G, u P is eulerlan, PeP then there must be a path P of length two in P with all three points of P in the same component of G. Thus removal of P from H yields a seml-eulerian graph and + s (G) m(@(G),G) THEOREM 6. (DISCONNECTED SUBSEMI-EULERIAN GRAPHS) Every disconnected graph + is subsemi-eulerian.If we set m(@(G),G) 0 when (G) , then s (G) 1 max P0 + k i m(@(G),G) 2).e THEOREM i. (SUBEULERIAN MULTIGRAPHS) Every noneulerian multigraph is sub- .ulerian.
i s+(M) -> e+(M) i p.. + k I e and we may state
Figure i Figure 4