International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 85-91 © TJPRC Pvt. Ltd.,
STRONG (WEAK) MATCHINGS & EDGE COVERINGS OF A GRAPH 1 1 2
R.S.BHAT, 2S.S.KAMATH & 3SUREKHA BHAT
Department of Mathematics, Manipal Institute of Technology, Manipal, 576 104
Dept of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal 575 025, Mangalore-India 3
Depatment of Mathematics, Milagres College, Kallianpur, Udupi
ABSTRACT Let G = (V, X) be a graph without isolates. Let de(x) denote the edge degree of an edge. An edge x=uv strongly (weakly) covers a vertex v if de(x) ≥ de(y) (de(x)≤ de(y)) for every edge y incident on v. A set L ⊆ X is a strong edge covering (SEC) [(weak edge covering (WEC)] if every vertex in G is strongly (weakly) covered by some edge in L. The strong edge covering number sα1 = sα1(G) (weak edge covering number wα1 = wα1(G)) is the cardinality of a minimum SEC (WEC) of G. An edge x is said to be strong (weak) if de (x) ≥ de(y) (de(x) ≤ de(y) ) for every y adjacent to x. A set L ⊆ X is a matching if no two edges in L are adjacent. A matching L of edges is a strong matching (SM) [weak matching (WM)] if every edge in L is strong (weak). The strong matching number sβ1 = sβ1(G) (Weak matching number wβ1 = wβ1(G)) is the cardinality of maximum SM (WM). In this paper, analogues to the Gallai’s theorem we prove that for any isolate free graph G with p vertices sα1 + sβ1 = wα1 + w β1 = p. Further it is proved that the following inequality chain holds in any graph. s β1 ≤ wβ1 ≤ β1 ≤ α1 ≤ wα1 ≤ sα1
KEY WORDS: Strong (Weak) Matchings, Strong (Weak) Edge Coverings. Subject Classification:
05C69
INTRODUCTION Throughout in this paper we consider simple, undirected, isolate free graph. For any undefined terminologies we refer [7]. The strong (weak) domination was first introduced by Sampathkumar and Pushpalatha [14]. For any two adjacent vertices u and v in a graph G = (V, X), u strongly (weakly) dominates v if d(u) ≥ d(v) (d(u) ≤ d(v)). A set D ⊆ V is a dominating set (strong dominating set [SDset], weak dominating set [WD-set] respectively) of G if every v∈ V - D is dominated (strongly dominated, weakly dominated respectively) by some u∈D. The domination number γ (G) (strong domination number γs (G), weak domination number γw (G) respectively)
is the cardinality of a
minimum dominating set (SD- set, WD-set respectively) of G. The domination concept is well studied in [10]. The strong domination has been further discussed in [1, 2, 3, 4, 5, 8, 9, and 12]. Geetha S.Naik [6] defined the strong (weak) edge domination numbers of a graph. A set L ⊆ X of a graph G is an edge
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dominating set if every edge in X-L is adjacent to some edge in L. The edge domination number γ ' = γ ' (G) of G is the cardinality of a minimum edge dominating set of G. Edge degree of an edge de(x), x∈X is defined as the number of edges adjacent to x in G. For any two adjacent edges x and y in G we say that x strongly (weakly) dominates y if de(x) ≥ de ( y) (de (x) ≤ de (y). A set L ⊆ X of a graph G is a strong edge dominating set (SED –set) [weak edge dominating set (WED-set)] if every edge in X-L is strongly (weakly) dominated by some edge in L. The strong edge domination number γs' = γs' (G) (γw' = γw' (G)) of G is the cardinality of a minimum SED-set (WED-set) of G. S.S. Kamath and R.S. Bhat [11] defined strong weak vertex coverings. If an edge is incident on a vertex we say that the vertex and the edge cover each other. For an edge x = uv, v strongly covers the edge x if d (v) ≥ d (u) in G. Then u weakly covers x. A set S ⊆ V (WVC)]
is a Strong Vertex Cover (SVC) [Weak Vertex Cover
if every edge in G is strongly [weakly] covered by some vertex in S. The strong vertex
covering number sαo(G) [weak vertex covering number wαo(G)] is the cardinality of a minimum SVC [ WVC]. A set L ⊆ X
is an edge covering (EC) if the edges of L cover all the vertices of G. Edge
covering number α1 = α1(G) is the cardinality of a minimum edge cover of G. We now define the edge analogue of strong (weak) vertex coverings as follows.
STRONG (WEAK) EDGE COVERING An edge x strongly (weakly) covers a vertex v if de(x) ≥ de(y) (de(x) ≤ de(y)), for every edge y incident on v. A set L ⊆ X is said to be a strong edge covering (SEC) [(weak edge covering (WEC)] if every vertex in G is strongly (weakly) covered by some edge in L. The strong edge covering number sα1 = sα1(G) (weak edge covering number wα1 = wα1(G)) is the cardinality of a minimum SEC (WEC) of G. A minimum SEC (WEC) is written as sα1 -set (wα1 -set). In what follows for L ⊆ X, by
V( L), we mean the set of all vertices incident on the edge set L.
STRONG (WEAK) FULL EDGE COVERING A set L ⊂ X is said to be Full Edge Covering (FEC) if every v ∈ V(L ) is covered by an edge in X – L. The full edge covering number fα = fα (G) is the cardinality of a maximum FEC of G. A set L⊂ X is said to be Strong Full Edge Covering (SFEC) [Weak Full Edge Covering (WFEC)] if every v∈V (L) is weakly [strongly] covered by an edge in X – L. The strong full edge covering number fsα= fsα(G) (weak full edge covering number fwα= fwα(G)) is the cardinality of a maximum SFEC (WFEC) of G. A maximum FEC is written as fα-set. A maximum SFEC (WFEC) is written as fsα-set (fwα-set).
STRONG (WEAK) MATCHINGS An edge x is said to be strong (weak) if de(x) ≥ de(y) (de(x) ≤ de(y)), for every y∈N(x). A set L⊆ X is said to be a matching if no two edges are adjacent. A matching L of edges is said to be a strong matching (SM) [weak matching (WM)] if every edge in L is strong [weak]. The strong (weak) matching number sβ1 = sβ1(G) (wβ1 = wβ1(G) is the cardinality of a maximum SM (WM). A maximum Strong (weak) matching is written as sβ1-set (wβ1- set).
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Strong (Weak) Matchings & Edge Coverings of a Graph
STRONG (WEAK) FULL MATCHINGS A set L⊂ X is said to be a cut set if G – L is disconnected. A cutest L is said to be a full matching if every component of G – L is an edge of G. The full matching number
fβ = fβ(G) is the
cardinality of a minimum full matching. A cut set L is a Strong Full Matching (SFM) [Weak Full Matching (WFM)] if every component of G – L is a strong [weak] edge of G. The strong full matching number fsβ = fsβ (G) (weak full matching number fwβ =fwβ (G)) is the cardinality of a minimum SFM (WFM).
ILLUSTRATIONS
x1
x2
x3
x4
x5
x6
x8
x7
G3
G1
G2 Fig.1
In G1 of Fig.1, α1-set is L1 = {x1, x3, x5, x6}, sα1-set is L2 = {x1, x7, x3 , x4, x8 , x6}, wα1-set is L3 = {x1, x2, x4,x5, x6}. Then X – L1 is a fα -set. X – L3 is a fsα -set where as X – L2 is a fwα -set. β1 -set is {x1, x3, x5}, sβ1-set is {x7}, wβ1-set is {x1, x6}. fβ – set is {x1, x7, x3 , x5, x8 } fsβ - set is X-{x7}, and fwβ –set is X -{x1, x6}. Here sβ1 = 1, wβ1 = 2, β1 = 3, α1 = 4 , wα1 = 5, sα1 = 6.and fα = 4 , fsα = 3 , fwα = 2 and fβ = 5, fwβ = 6 and fsβ = 7. In a rooted tree an edge x is called a root edge if x is incident on the root of the tree. In any tree an edge adjacent to a pendant edge is called a support edge. In G2 of Fig 1, set of all pendant edges together with the root edges is both α1 –set as well as wα1-set. Set of all edges except one of the root edges forms a sα1-set. Set of all pendant edges together with one of the root edges forms β1 as well as wβ1-set, where as any two independent support edges form a sβ1-set. Set of all support edges form fα set as well as fsα -set. The singleton set containing one of the root edges form fwα -set. We observe the following facts. Fact 1. A graph G is said to be edge regular if all the edges are of same edge degree. For any edge regular graph α1 = sα1= wα1 and fα = fwα = fsα and sβ1 = wβ1 = β1. But the converse is not true. The graph G3 in Fig.1, is an example of a graph which is not edge regular but all the parameters are equal. Here α1 = sα1 = wα1 = 3 and fα = fwα = fsα = 4 and β1 = sβ1 = wβ1 = 2. Note that even though the value of all the parameters are equal the corresponding optimum sets are not equal. Fact 2. From the definition of matching number and full matching number it is easy to observe that β1 + fβ = q. Similarly we have sβ1 + f sβ = q and wβ1 + f wβ = q
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Fact 3. The full edge covering parameters need not be defined for all graphs. Whenever it is not defined we take the value of the parameter to be zero. For example for the star K1,n ;
fα = fwα = fsα = 0.
Further any pendant edge cannot be in a FEC as every edge in a FEC is adjacent to at least one edge at both the end vertices. For the similar reason no pendant edge exists in a SFEC or WFEC.
MAIN RESULTS The following result gives a necessary and sufficient condition for a set L ⊂ X to be a FEC (SFEC, WFEC). Proposition 2.1. Let G = (V, X) be an isolate free graph and L ⊂ X. Then i)
L is a FEC if and only if for every y = uv in L there exist at least two edges x and z in X - L such that x is incident on u and z is incident on v.
ii)
L is a SFEC (WFEC) if and only if for every edge y = uv in L there exist at least two edges x and z in X - L such that x is incident on u and z is incident on v with de(y) ≥ max{de(x), de(z)} [de(y) ≤ min{de(x),de(z)].
Proof. (i) Let L ⊂ X be a full edge covering of G. Let y∈ L and y = uv. By definition of FEC, every v ∈V(L) is covered by some edge in X – L. Hence u is covered by some edge say x ∈X- L and v is covered by some edge say z∈ X - L. Thus condition in the theorem is satisfied. Converse is trivial. (ii) If L is a SFEC then L is a full edge cover and by definition of SFEC, every v ∈V(L) is strongly covered by some edge in X - L. Let y ∈L and y = uv. Then each of u and v are weakly covered by an edge in X – L. This implies that there exist at least two edges
x = us and z = vt
in X – L such that de
(y) ≥ de(x) and de( y) ≥ de(z). Therefore de (y) ≥ max { de(x), de(z)}as desired. The argument in respect of WFEC is similar and hence we omit the proof. Conversely, if every edge in L satisfy the condition, then it is not hard to verify that L is a full edge covering. It is well known from Gallai’s Theorem that α1 + β1 = p. Analogously we prove the following. Our proof is similar to the proof of Gallai. Proposition 2.2. For any isolate free graph G = (V, X) with p vertices, sα1 + sβ1
= p
(1)
wα1 + w β1 = p
(2)
Proof. Let M1 be a sβ1-set so that │M1│= sβ1. We construct a SEC, Y using M1. Let X1 be the set of edges obtained by choosing one edge x incident on each unsaturated vertex v such that de (x) ≥ de(y) for every y incident on v. Then Y = M1 ∪ X1 is a strong edge covering. Since │M1│+│Y│≤ p and │Y│≥ sα1 we have sα1 + sβ1 ≤
p. On the other hand if L is a SEC we can get a strong matching of size
p − L by selecting strong edges which are independent in L. Therefore sβ1 ≥ p - sα1. Hence the result (1) follows. With the similar argument we can prove (2).
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Strong (Weak) Matchings & Edge Coverings of a Graph
Proposition 2.3. Let G = (V, X) be a graph without isolates and L ⊂ X. Then, i.
L is a FEC if and only if X-L is an EC.
ii.
L is SFEC if and only if X-L is a WEC.
iii.
L is WFEC if and only if X-L is a SEC.
Proof. We prove (i) only. (ii) and (iii) can be proved with similar argument. Let L be a FEC. Partition the vertex set V in to two sets V1 and V2 such that V1 = V(L) and V2 = V – V1. From the definition of FEC it is clear that every vertex in V1 is covered by an edge in X – L. If w∈ V2 then w is an end vertex of some edge in X – L. Therefore every vertex in G is covered by some edge in X – L and hence X - L is an edge covering of G. Converse is trivial. Proposition 2.4. Let G =(V, X) be a graph without isolates, with p vertices and q edges. Then,
α1 + fα = q
(3)
sα1 + fwα = q
(4)
wα1 + fsα = q
(5)
Proof.
Let L be an α1 –set. Then L is an EC and then X - L is a FEC by Proposition 2.3.
Hence
f α ≥ X − L ≥ q − α 1 ….(i). On the other hand let T be a fα-set. Then T is a FEC and hence X
- T is an EC by Proposition 2.3 . Therefore
α1 ≤ X − T ≤ q − fα .
Hence
f α ≤ q − α1
….(ii).
Then (3) follows from (i ) and (ii). With the similar argument we can prove (4) and (5). Proposition 2.5. Let G be an isolate free graph with p vertices and q edges. (i) (ii)
If there exists an EC, which is also a FEC then
α1 ≤
q and 2
If there exists a SEC (WEC) which is also a SFEC ( WFEC ) Then sα1 + wα1 ≤ q.
Proof. (i) Let L be an EC which is also a FEC. Since L is an EC we have
α1 ≤ L
. Also since L is a
FEC we have from Proposition 2.3, X - L is an EC. Hence we have α1≤│X - L│= q -│L│. Adding the two results we get
α1 ≤
q . 2
(ii) Let L be a SEC which is also a SFEC. Since L is a SEC we have
sα 1 ≤ L . Also since L is a
SFEC, X - L is a WEC by Proposition 2.3. Hence we have wα1 ≤ │X - L│= q -│L│. Adding we get sα1 + wα1 ≤ q. Then fwα + fsα ≥ q follows from (4) and (5). In the next result we get a relationship between the newly defined parameters.
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R.S.Bhat,S.S.Kamath & Surekha Bhat
Proposition 2.6. The following inequality chain holds in any isolate free graph G, sβ1 ≤ wβ1 ≤ β1 ≤ α1 ≤ wα1 ≤ sα1
(6)
fwα ≤ fsα ≤ fα ≤ fβ ≤ fwβ ≤ fsβ Proof. First we show that β1 ≤ α1 . It is well known that β1 ≤
≥
(7)
p . Then by Gallai’s theorem we have α1 2
p p . Therefore β1 ≤ ≤ α1. As every WEC is an EC we have α1 ≤ wα1. To show that wα1 ≤ sα1 : 2 2
Let wα1 = k and W = { x1, x2, … , xk} be a wα1 – set. Without loss of generality we assume that de(x1) ≤ de (x2) ≤…
≤ de (xk). Choose y1∈ N [x1] such that de(y1) is as big as possible. S1 = {y1}. Now for
each xi ∈W choose yi ∈N[xi] – Si – 1,
2 ≤ i ≤ k, such that de (yi) is as big as possible where Si = Si - 1∪{yi}.
Then Sk = Sk – 1 ∪ {xk}= { y1,y2, …. ,yk}. By the choice of the edges, each yi is distinct and uniquely strongly cover some vertex in V. Therefore Sk ⊆ S for some sα1 –set S. Hence wα1 =│W│=│Sk│≤ │S│= sα1 as desired. To show that sβ1 ≤ wβ1 ≤ β1 : We have now shown that α1 ≤ wα1 ≤ sα1. Then by Gallai’s theorem and Proposition 2.4, we have p- β1 ≤ p -w β1 ≤ p - s β1 which yields the desired result. Using Proposition 4, equation (6) can now be written as q - fsβ ≤ q - fwβ ≤ q - fβ ≤ q - fα
≤ q - fsα
≤ q – f wα
which yields
(7). We observe that the graph G1 in Fig. 1 above is an example of the graph where the strict inequality in (6) and (7) holds. Corollary 2.6.1. The following inequality chain holds in any isolate free graph G, sβ1 ≤ wβ1 ≤ β1 ≤ α o ≤ wα o ≤ sα o
(8)
sβo ≤ wβo ≤ βo ≤ α1 ≤ wα1 ≤ sα1
(9)
Proof. We have α o ≤ wα o ≤ sα o and sβo ≤ wβo ≤ βo (See [8] ). Also we have β1 ≤ α o. and then by Gallai’s theorem we have βo ≤ α1 .Then (8) and (9) follow from (6).
We now relate edge domination, strong edge domination and strong covering numbers. Proposition 2.7. Let G = (V,X) be a graph without isolates. Then γ' ≤ γs' ≤ sα1
(10)
γ' ≤ γw' ≤ wα1
(11)
Proof. (10) follows from the fact that any SEC is a sed- set and any sed – set is an edge dominating set. Similarly any WEC is a wed set and any wed set is an edge dominating set. Hence (11) follows.
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