Total Matchings and Total Coverings of Graphs

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Y . Alavi

M. Behzad

L. M. Lesniak-Foster

E. A. Nordhaus

WESTERN MlCHlGAN UNlVERSlTY

ARYA-MEHR UNIVERS1TY

LOUlSlANA SJA JE UNlVERSlTY

MlCHlGAN SJAJE UNlVERSlJY

ABSTRACT

In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.

1. PRELIMINARIES

Let G be a finite undirected graph with no loops or multiple edges

Journal of Graph Theory, Vot. 1 (1977) 135-140 Copyright @ 1977 by John Wiley & Sons, Inc. 135

136 JOURNAL OF GRAPH THEORY

having vertex set V(G) and edge set E (G) . The elements of V(G)U E ( G ) are called elemenrs of G. A vertex u of G is said to cover itself, all edges incident to u, and all vertices adjacent to u. Similarly, an edge e of G covers itself, the two end vertices of e, and all edges adjacent to e. Two elements of G are called independent if neither one covers the other. A set C of elements of G is called a toral couer if the elements of C cover all elements of G and C is minimal. A set M of elements of G is called a total matching if the elements of M are pairwise independent and M is maximal. For a given graph G, let a2(G) = inf ICI and a;(G) = sup ICI, where the inf and the sup are taken over all total covers C of G. Similarly let P;(G)=inf [MI and P2(G)=supIMI, where the inf and the sup are taken over all total matchings M of G. Related ideas are considered in [2, 3, 6, 8, 91. For additional graph theory terminology, see [I] and [7].

2. GENERAL RESULTS

Let G be a connected graph and M a total matching for G. Since M is maximal, every element of G not in M must be covered by some element of M, and since every element of M covers itself and is covered by no other element of M, the set M is a total cover. An immediate consequ- ence of this observation is the following chain of, inequalities.

THEOREM I. For every connected graph G,

We proceed to develop upper and lower bounds for the four parameters we have formulated. In Table I , precise values of the parameters will be given for certain special classes of graphs, which in turn will establish the

TABLE I

Graphs

para- ( p > l ) K,,, ( 2 s m = n f b 2 3 ) G connected G KP Km." PP CP

meters (complete) (star) (Bi-partite) (path) (cycle) Tz ( p t l )

TOTAL MATCHINGS AND TOTAL COVERINGS OF GRAPHS 137

sharpness of all but one of the bounds. In the discussion which follows, [XI and {x} denote, respectively, the greatest and the least integer functions.

THEOREM 2. If G is a connected graph of order p B 2, then

{ p/2} d CIS( G) 5 p - 1.

Proof. To establish the lower bound, let C be any total cover of G all of whose elements are edges. Since an edge covers exactly two vertices, a ( G ) B I C I L { p/2}.

We next proceed to establish the upper bound. Since G is connected and p Z 2 , G has at least one edge. Let C be a total cover for G of maximum cardinality, and suppose that ICl= a;(G) B p . We seek a con- tradiction. Let C = V1 U El, where V1 is the set of vertices and El the set of edges in C. Then El is not empty, since otherwise C contains all p vertices of G and is clearly not minimal. The set V1 is also nonempty, for otherwise the induced graph (El) is a spanning subgraph of G with at least one cycle. But then if e is any edge of this cycle, the set El-{e} covers all elements of G, which contradicts the minimality of C.

Since C is minimal, no edge of El is incident with two vertices of Vl. The edges of El therefore fall into two sets E ; and E i , where the edges of E: have one end vertex in V1 and one in Vz= V(G)- V1, and the edges of E ; have both end vertices in V,.

If e = ulvz is an edge of E: with v I E V1 and U,E V,, then no edge of E ; can have an end vertex at uz, since the edge e could then be deleted from C to obtain a total covering set C', contradicting the minimality of C. Let V3 be the set of vertices in Vz incident with at least one edge of E; . It is impossible for two edges of E; to have a vertex of V3 in common, since one could delete one of these edges from C and obtain a set covering all elements of G.

Each vertex of V3 is therefore incident with exactly one edge of E ; , SO lE{ (= IV31. Let V4= V2- V3. If V4 is assumed to be empty, then

I CI = I ViI+ IEI I = I Vi( + 1 V,J = I V(G)I = P ,

However, C is then not minimal, since at least one vertex in V1 which is an end vertex of an edge in E ; could be deleted from C, and the resulting set of elements is a total covering set of G. Thus V, is not empty. Let H denote the induced graph (VJ. The set E ; covers all edges and some (possibly all) vertices of H. Let el, e2, . . . , e,, where t L 1, be the edges of Eq. Then el is incident with two vertices of H. The edges el and ez cover at least three vertices of H, and since no three edges of E ; form the edges

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of a cycle C2 or a path P4, then E’; has at most lV41- 1 edges. Therefore IC/ 5 I V1l + I V31 + I V,l- 1 5 p - 1, a contradiction. I

The authors wish to thank the referee for suggesting a shorter proof of the next theorem.

THEOREM 3. If G is a connected graph of order p, then

1 SaZ(G)S{p/2}

Proof. The lower bound is obvious. To establish the upper bound, we employ induction on connected graphs of order p, and readily verify the theorem for p S 3 . Let vertices u l , u 2 , . . . , u , , - ~ , u,, define a longest path in G. Two cases arise: Case (1). If no end vertex is adjacent to u , , - ~ , the graph obtained by removing from G the vertices u , - ~ , u,,, and all edges incident to them is connected, and can therefore be covered by {p/2}- 1 or fewer elements. Those removed can be covered by the edge u,,-lu,,. Case (2). If at least one end vertex is adjacent to u , , - ~ , we remove from G the vertex u , , - ~ , all edges incident to u , , - ~ , and all end vertices adjacent to u , , - ~ . The resulting graph is connected or empty, and can be covered by {p/2}-1 or fewer elements. Those removed can be covered by vertex U , - l . 1

THEOREM 4. If G is a connected graph of order p L 2 , then

1 5 & ( G ) 5 p - 1 and {p/2}5 &(G) 5 p - 1.

Proof. The bound 1 5 Pi(G) is obvious, and by Theorems 1 and 2, /3;( G) d p2( G ) S a;( G ) 5 p - 1. The remaining bound { p / 2 } I p2( G) will be established as a corollary of Theorem 5. The upper bound for P;(G) may not be sharp. I

3. EVALUATION OF PARAMETERS FOR SPECIAL GRAPHS

By using techniques and arguments similar to those employed in the proofs of the theorems of Sections 2 and 4, together with the bounds obtained in Section 2 , one can evaluate the parameters a2(G), ai (G) , P2(G), P;(G) for many classes of graphs. Several such results are sum- marized in Table I. In this table, T; denotes a tree of order p = 2n + 1 whose n end vertices are each at distance two from the center vertex. An examination of the table shows that, with the exception of the upper bound for P;(G), all bounds obtained in Section 2 are sharp.

TOTAL MATCHING AND TOTAL COVERING OF GRAPHS 139

4. VERTEX AND EDGE COVERING AND INDEPENDENCE NUMBERS

For a given graph G, let ao(G) = inf ICol and ab(G) = sup lCol, where the inf and sup are taken over all total covers Co satisfying C0s V(G), and let Po(G) =sup lMol and P&(G) = inf IMol, where the inf and sup are taken over all total matchings Mo for which MOC V(G).

Similarly, let a l ( G ) = inf IC,J and a;(G) = sup IC1l, where the inf and sup are taken over all total covers C1 satisfying C l c E ( G ) , and let Pl(G) = sup (Mlj and P;(G) = inf IM1l, where the inf and sup are taken over all total matchings M1 for which MI E E ( G ) . Some special values of P{(G) have been noted by Grunbaum [ 5 ] .

As a result of these definitions for covers one has the inequalities az(G) 5 a i (G)S al(G) 5 cr$(G) for i = 0 or 1. Similarly, for matchings one has P2(G)2zmax(Po(G), P,(G)) and it can be shown that P$ (G)Z

A well known result due to Gallai [4] states that a graph G of order p having no isolated vertices satisfies ao(G)+ P0(G) = crl(G) + Pl(G) = p. An extension of Gallai's result was made by Meng [9], who showed that ab(G)+Pb(G)=p. Theorem 5 presents bounds for the sum a2(G)+ &(G). This sum can exceed p , as shown by the complete graph G = Kp, p odd.

max (Pb(G), P;(G)).

THEOREM 5. If G is a connected graph of order p 2 2, then p 5 az(G)+ Pz(G) d { 3 p / 2 } - 1.

Proof. To establish the lower bound, we first show that there is a total cover T of G with cardinality a2(G) such that the edges in T are pairwise nonadjacent. Let T1 be a total cover of G with cardinality az(G). Suppose T1 contains adjacent edges of G. Let e = vu and f = uw be two such edges. Then T2 = ( T , -{f)) U {w} is a total cover of G. Moreover, IT21=IT11 and T2 contains at least one less pair of adjacent edges of G than does Tl. By continuing this process (if necessary), we obtain a total cover T of G with the desired properties.

Let V* denote the set of vertices in T and let E* denote the set of edges in T. Furthermore, let n = IV*( and m = IE*J. Consider the subset V' of V(G) consisting of all vertices of G which are neither in V* nor incident with an edge in E*. We wish to show that V 'UE* is indepen- dent. By the way in which V ' U E* was chosen, no two edges in V'U E* are adjacent and no vertex in V' U E* is incident with an edge in V ' U E*. Thus we need only show that if ul, u2e u', then u l u 2 & E(G). Suppose, to the contrary, that u1v2 E E ( G). Since v1 and v2 are in V', (i) neither VI nor

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u2 is in V* and (ii) neither u1 nor u2 is incident with an edge in E*. Therefore no element of T covers the edge u1u2, which contradicts the fact that T is a total cover of G. Thus u 1 u 2 & E ( G ) and V 'UE* is independent. We conclude that &(G) 2 1 V' U E*[ z ( p - n - 2m) + m = p-cu,(G). The lower bound given in Theorem 5 is sharp, since equality holds for a star graph G of order p L 2 .

If the upper bound for a2(G) given in Theorem 3 is combined with the upper bound for P2(G) given in Theorem 4, then

~ z ( G ) + @ z ( G ) ~ { ~ ~ / ~ ) - 1.

This upper bound may not be sharp. I

Corollary 5.1. If G is a connected graph of order p , then

&(G) 2 (~121. Proof. By Theorems 1 and 5 , 2P2(G) B a2(G)+ &(G) 2 p. I

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[4] T. Gallai, Uber extreme Punct und Kantenmengen, Ann. Univ. Sci.

[ 5 ] B. Grunbaum, Matchings in polytopal graphs. Networks 4 (1974)

[6] R. P. Gupta, Independence and covering numbers of line graphs and total graphs, Proof Techniques in Graph Theory. Academic Press, New York (1969) 61-62.

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[7] F. Harary, Graph Theory. Addison-Wesley, Reading, MA. (1969). [S] A. Meir and J. W. Moon, Relations between packing and covering

[9] D. Meng, Matchings and coverings for graphs, Ph.D. thesis, Michigan numbers of a tree. Pac. J. Math. (to appear).

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