Convexity of minimal total dominating functions in graphs

Convexity of Minimal TotalDominating Functionsin Graphs

Bo Yu*†DEPARTMENT OF COMBINATORICS AND OPTIMIZATION

UNIVERSITY OF WATERLOO, WATERLOO, ONTARIOCANADA, N2L 3G1

E-mail: [email protected]

ABSTRACT

A total dominating function (TDF) of a graph G = (V,E) is a function f : V → [0, 1] suchthat for each v ∈ V , the sum of f values over the open neighbourhood of v is at leastone. Zero-one valued TDFs are precisely the characteristic functions of total dominatingsets of G. We study the convexity of minimal total dominating functions. A minimaltotal dominating function (MTDF) f is called universal if convex combinations of f andany other MTDF are minimal. Generalizing and unifying two previous major results byCockayne, Mynhardt and Yu in the area, we give a stronger sufficiency condition for anMTDF to be universal. Moreover, we define a splitting operation on a graph G, whichpreserves the universality. Using the operation, we give many more classes of graphshaving a universal MTDF. c© 1997 John Wiley & Sons, Inc.

1. INTRODUCTION

In this paper, we will only consider finite, simple undirected graphs without isolated vertices.A total dominating set of a graph G = (V,E) is a subset X of vertices such that every vertexof G is adjacent to at least one vertex in X . The smallest such set is called a minimum totaldominating set. Much research has been devoted to the study of minimum total dominating

*The work was partially supported by the National Sciences and Engineering Research Council ofCanada, under Grant 156585.† A Preliminary version of this paper has appeared in: ''Convexity of minimal total dominatingfunctions in graphs,'' Proc. Computing and Combinatorics, First Annual International Conference,COCOON'95, LNCS 959, Springer, New York, (1995), pp. 357–365.

Journal of Graph Theory Vol. 24, No. 4, 313 321 (1997)c© 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/040313-09

314 JOURNAL OF GRAPH THEORY

sets and related topics. The reader is referred to Hedetniemi and Laskar [7] for an excellentbibliography concerning domination in graphs.

In [9], Laskar et al. showed the problem of finding a minimum total dominating set to be NP-hard. The problem may be formulated as an integer programming problem. (IP): min

∑v∈V f(v)

subject to∑

u∈N(v) f(u) ≥ 1 and f(v) ∈ {0, 1} for each v ∈ V , where N(v) is the openneighbourhood of a vertex v, defined as the set {w ∈ V |wv ∈ E}. It is worth noting that Domkeet al. [4], Hedetniemi et al. [6], and Laskar et al. [8] have studied the linear relaxation of a similarproblem with N(v) replaced by N [v], where N [v] = N(v) ∪ {v}. In this paper we study thelinear relaxation of (IP), i.e., we replace the constraint f(v) ∈ {0, 1} by 0 ≤ f(v) ≤ 1 for eachv ∈ V . This motivates the following definition. A total dominating function (TDF) of the graphG = (V,E) is a feasible solution to the linear programming relaxation of (IP). Zero-one valuedTDFs are precisely the characteristic functions of total dominating sets of G. A minimal totaldominating function (MTDF) f is a TDF such that f does not remain a TDF if for any v ∈ V ,the value f(v) is decreased. Note that an optimal solution to (IP) or to its linear programmingrelaxation must be an MTDF. The main goal of this paper is to study the convexity of MTDFs.The motivation of studying the convexity of minimal total dominating functions is as follows. Itis easily shown that a convex combination of two TDFs is still a TDF. But a convex combinationof two MTDFs is not necessarily an MTDF (see Figure 1 in Section 2). Motivated by this fact,Cockayne et al. [2] introduced the notion of a universal MTDF. An MTDF g is called universalif any convex combination of g and any other MTDF is also an MTDF. Another motivation forstudying the convexity of MTDFs is the following interpolation problem raised by Hedetniemi[5] (the interpolation problem in [5] concerns dominating functions): The aggregate is an MTDFf is the sum of f values over all vertices. Suppose that we are given two MTDFs f and g withaggregates α and β, and a number t ∈ (α, β). Does G have an MTDF with aggregate t? Itis easily seen that the answer to the interpolation problem is ‘‘yes’’ provided G has a universalMTDF. The aggregate of an MTDF is an important parameter of a graph, which has a close relationwith other well-studied parameters, such as the domination number, the irredundance number, thevertex independence number, the total domination number, and the fractional domination number;see [4, 8].

Characterizations of universal MTDFs and graphs with universal MTDFs are studied in [1, 2,3, 11]. At present, there are two major results in this area. Cockayne et al. [2] gave a sufficiencycondition for an MTDF to be universal in general graphs, and Cockayne and Mynhardt [1] gavea necessary and sufficient condition for an MTDF to be universal in trees.

The main contributions of this paper are a new sufficiency condition for an MTDF to beuniversal and the introduction of an operation on graphs, called ‘‘splitting,’’ which preserves theuniversality of MTDFs. In Section 2, we give the new sufficiency condition for an MTDF to beuniversal in general graphs. This result generalizes and unifies the two previous major results;see [1, 2]. It is possible that our condition is also necessary for an MTDF to be universal. Thesecond main result, in Section 3, is an operation on graphs, called the splitting operation. A vertexv in a graph G is ‘‘split’’ if we add a new vertex w to G and join w with every neighbour of v.We show that a graph G has a universal MTDF if and only if the graph obtained by applying thesplitting operation to G has a universal MTDF. This operation gives many more classes of graphsthat have universal MTDFs. Finally, in Section 4, we list some open problems.

2. A SUFFICIENT CONDITION FOR AN MTDF TO BE UNIVERSAL

In this section, we introduce two types of vertices, dominating vertices and low vertices, whichplay an important role in characterizing universal MTDFs. In a tree, dominating vertices and

CONVEXITY OF MINIMAL TOTAL DOMINATING FUNCTIONS 315

FIGURE 1. A convex combination of f and g is not an MTDF.

low vertices are equivalent to so-called short vertices and hot vertices defined in [2], respectively.Our main theorem, Theorem 3, gives a sufficient condition for an MTDF to be universal. Oursufficiency condition is stronger than the one in one of the main theorems (Theorem 18) in [2].Our condition shows that the MTDF in Example 1 is universal, but this MTDF does not satisfythe sufficiency condition in Theorem 18 in [2]. We have not succeeded in finding an exampleof a universal MTDF that does not satisfy our sufficiency condition. If we restrict the graph inTheorem 3 to be a tree, then the condition in our theorem is necessary and sufficient; see [1]. Weconclude this section with two examples.

For a TDF f , denote∑

u∈N(v) f(u) by f(N(v)). Let Bf be the set {v ∈ V |f(N(v)) = 1},called the boundary of f . Let Pf be the set {v ∈ V |f(v) > 0}. For subsets A,B of V , we writeA → B if every vertex in B is adjacent to some vertex in A, i.e., B is a subset of ∪v∈AN(v).Recall that a TDF f is minimal if f does not remain a TDF when the value f(v) is decreasedfor any v ∈ V . For two MTDFs f and g, we define a convex combination of f and g to behλ = λf + (1 − λ)g, where 0 < λ < 1. Note that λ is not allowed to be 0 or 1. We need thefollowing two theorems, Theorem 1 and Theorem 2, from [2].

Theorem 1 [2]. A TDF f is minimal if and only if Bf → Pf .

The following theorem gives a necessary and sufficient condition for a convex combination oftwo MTDFs to be an MTDF.

Theorem 2 [2]. Let f and g be MTDFs. A convex combination of f and g, hλ = λf+(1−λ)g,where 0 < λ < 1, is an MTDF if and only if Bf ∩Bg → Pf ∪ Pg .

The above theorem shows that the minimality of hλ is independent of λ; that is, either allconvex combinations of f and g are minimal or none are minimal. Using this theorem, it is easilyshown that in Figure 1 convex combinations of f and g are not minimal.

Recall that an MTDF g is called universal if any convex combination of g and any other MTDFis also an MTDF. From a geometric point of view, if we denote the set of all MTDFs of G by F ,then, in the terminology of the literature on convex sets [10, pp. 4–5], g is a universal MTDF ifand only if F is star-shaped relative to g. The set of universal MTDFs is the kernel of the set F ,and it follows that the set of universal MTDFs of a graph of a convex set (see Theorem 1.2 [10]),i.e., a convex combination of two universal MTDFs is also a universal MTDF.

A vertex v is called a dominating vertex if there exists a vertex u such that the open neigh-bourhood of u is properly contained in the open neighbourhood of v, i.e., N(u) ⊂ N(v), where


FIGURE 2.

‘‘⊂’’ means the relation of proper containment. We denote the set of dominating vertices of thegraph G by D(G).

Proposition 1. For an MTDF f of G and a dominating vertex v in the boundary of f , ifu ∈ V (G) has N(u) ⊂ N(v), then

(1) u is in the boundary of f , and(2) f(w) = 0 for all w ∈ N(v) \N(u).

Proof. Obvious.

Now we introduce a type of vertex that is important in our study of universal MTDFs. Vertexv is called f -low if there exists an MTDF f such that Bf ∩N(v) ⊆ D(G). An f -low vertex vmay have Bf ∩N(v) empty. Further, v is called low if it is f -low for some MTDF f .

Proposition 2. If f is an MTDF and vertex v is f -low, then f(v) = 0.

Proof. First, we claim that if vertex v is f -low, then for any u ∈ Bf ∩ N(v), there exists avertex w such that N(w) ⊆ N(u) \ {v}.

Since v is f -low, by the definition, any vertex u ∈ Bf ∩N(v) is a dominating vertex. Thenthere exists a vertex x such that N(x) ⊂ N(u). If x 6∈ N(v), then we are done. Otherwise letx ∈ N(v). Since u ∈ Bf , Proposition 1 shows that x ∈ Bf . Since x ∈ Bf ∩N(v), x is also adominating vertex. We repeat the previous argument to get a sequence of vertices u, x, x1, . . .,such that N(u) ⊃ N(x) ⊃ N(x1) · · ·. See Figure 2. Since N(u) is finite, the sequence must befinite. Hence, there exists a vertex w such that N(w) ⊆ N(u) \ {v}. This proves the claim.

Now we use the claim to prove the proposition. If f(v) > 0, then there exists u ∈ Bf ∩N(v) ⊆ D(G). By the claim, there exists a vertex w such that N(w) ⊆ N(u) \ {v}. ByProposition 1, w ∈ Bf and f(v) = 0. This is a contradiction. Therefore f(v) = 0.

Before we state the main theorem of this section, we introduce another type of vertex. Let Wbe the subset of vertices v such that for each u ∈ N(v), there exists t in N(u) \ {v} with t ∈ Bf

for every MTDF f . We now state the sufficient condition for an MTDF to be universal.


FIGURE 3. g is universal.

Theorem 3. An MTDF g is a universal MTDF of G if

(i) all vertices, except possibly dominating vertices and vertices in W , are in the boundary ofg, and

(ii) all low vertices v have g(v) = 0.

Proof. Let f be any MTDF. By Theorem 2, we only need to show Bg ∩Bf → Pf ∪Pg . Letv ∈ Pf ∪ Pg . It suffices to show Bg ∩Bf → {v}. First, we claim that v is not f -low, for other-wise f(v) = 0 by Proposition 2 and g(v) = 0 by (ii), contradicting v ∈ Pf ∪ Pg . Therefore, bythe definition of low vertices, there exists a vertex x ∈ Bf ∩N(v) but x 6∈ D(G). If furthermorex 6∈ W , then x ∈ Bg by (i). It follows that x ∈ Bg ∩ Bf ∩ N(v). So Bg ∩ Bf → {v}. Nowwe may assume x ∈ W . Since v ∈ N(x), by the definition of W there must exist a vertext ∈ N(v) \ {x} such that t is in the boundary of every MTDF. Therefore t ∈ Bg ∩ Bf ∩N(v);that is Bg ∩Bf → {v}. Thus Bg ∩Bf → Pf ∪ Pg .

The following theorem states that the second condition of Theorem 3 is also a necessarycondition.

Theorem 4. If an MTDF g is universal, then g(v) = 0 for all low vertices v.

Proof. Let v be f -low for some MTDF f . Since g is universal, a convex combination hλ off and g is an MTDF, where hλ = λf + (1 − λ)g and 0 < λ < 1. Also Bhλ

= Bf ∩ Bg andPhλ

= Pf ∪ Pg . Hence, Bhλ∩N(v) = (Bf ∩Bg) ∩N(v) ⊆ Bf ∩N(v) ⊆ D(G). It follows

that v is hλ-low. Therefore hλ(v) = 0 by Proposition 2. This implies that g(v) = 0.

Notice that by the fact that short vertices [2] are a subset of dominating vertices and hot vertices[2] are a subset of low vertices, Theorem 3 together with Theorem 4 generalizes one of the maintheorems (Theorem 18) in [2]: An MTDF g is universal if all vertices, except possibly shortvertices, are in the boundary of g and all hot vertices have g value zero. Furthermore, Theorem 3is also a necessary condition when G is a tree (see [1]).

The following Example 1 comes from [2], where the MTDF g in Figure 3 does not satisfy thenecessary condition in [2], but was shown to be universal by long case analysis. We show that


FIGURE 4. A regular graph without a universal MTDF.

the MTDF g satisfies the condition in Theorem 3. Hence g is universal. Therefore our conditionis stronger. First, we give a necessary condition for a vertex to be low.

Proposition 3. If v is a low vertex, then every vertex u ∈ N(v) \ D(G) has degree atleast three.

Proof. Since v is f -low for some MTDF f, u 6∈ Bf for each u ∈ N(v) \ D(G). Sof(N(u)) > 1, i.e., f(v) +

∑w∈N(u)\{v} f(w) > 1. Since f(v) = 0 by Proposition 2, there are

at least two vertices x and z in N(u)\{v} such that f(x) > 0 and f(z) > 0. Hence u has degreeat least three.

Example 1 [2]. Consider the graph G and the MTDF g of Figure 3. The solid square verticesare the vertices of Bg = V \ {c}. D(G) = {c}.

By Proposition 3, vertices a, b and d are not low vertices. We now show that vertex c is notlow either. If vertex c is low, then there is an MTDF f such that Bf ∩ N(c) ⊆ D(G) = {c}.So Bf ∩ N(c) = ∅, i.e., vertices a, b and d are not in Bf . Since Bf → Pf by Theorem 1, wemust have f(x) = f(y) = f(c) = 0. It follows f(a) = 1 since f(a) = f(a) + f(y) + f(c) =f(N(d)) ≥ 1 and f(a) ≤ 1 by the definition of f . Hence f(N(d)) = f(a) = 1, i.e., vertex dis in Bf , a contradiction. Therefore vertex c is not low. g satisfies the condition (i) in Theorem3 since Bg = V \ {c}, indicated by the solid square vertices. Also, the condition (ii) followsfrom g(x) = g(y) = 0 since x and y are the only vertices that could possibly be low vertices.Therefore g is universal by Theorem 3.

Theorem 4 has an easy but useful corollary concerning graphs without universal MTDFs.

Corollary 1. If a graph G has a vertex v such that all its neighbours are low vertices, then Ghas no universal MTDF.

Proposition 4. If a regular graph has no low vertex, then it has a universal MTDF.

Proof. Suppose that the graph is r-regular. The function f having f(v) = 1/r for everyvertex v is an MTDF with Bf = V . Since the graph has no low vertex, f is universal byTheorem 3.


We close this section by showing a 3-regular graph that has no universal MTDF.

Example 2. Using Theorem 1, it is easy to check that g in Figure 4 is an MTDF. Note that Bg

consists of the solid square vertices. Notice that x is g-low since Bg ∩N(x) = ∅. Therefore, bysymmetry, y and z are also low vertices. Hence all neighbours of v are low vertices. By Corollary1, this graph has no universal MTDF.

3. THE SPLITTING OPERATION

In this section, we define an operation on a graph G that leads to a class of graphs, denotedby T (G), such that if any graph in T (G) has a universal MTDF, then all graphs in T (G) haveuniversal MTDFs. Let v ∈ V (G) be any vertex and w 6∈ V (G). Then v is split when we addw and the edges {wx|x ∈ N(v)} to G. Note that N(w) = N(v). Let the resulting graph bedenoted by G(v;w). The proof of the following theorem applies to 0–1 universal MTDFs too.This is stated in Corollary 3 below.

Theorem 5. G has a universal MTDF if and only if G(v;w) has a universal MTDF.

Proof. Let f be a universal MTDF ofG(v;w). Given f , define the function g : V (G) → [0, 1]to be:

g(t) =

{f(w) + f(v) if t = vf(t) otherwise.

To see that g(v) ≤ 1, suppose that f(v) > 0 or f(w) > 0. Then by Theorem 1 there is a vertexx ∈ N(v) (N(w) = N(v)) such that x ∈ Bf . Hence g(v) = f(w) + f(v) ≤ f(N(x)) = 1. ByTheorem 1, it is easy to see that g is an MTDF of G by checking Bg → Pg . Now we claim thatg is a universal MTDF of G. Let g1 be any MTDF of G. By Theorem 2, it suffices to show thatBg1

∩Bg → Pg1∪ Pg . First, extend g1 to be an MTDF f1 of G(v;w) by

f1(t) =

{0 if t = wg1(t) otherwise.

Since f is universal for G(v;w), we have Bf ∩ Bf1→ Pf ∪ Pf1

. Moreover, sinceN(w) = N(v), therefore (Bf ∩ Bf1

) \ {w} → Pf ∪ Pf1. From the construction, it follows

that Bg = Bf \ {w}, Pg = Pf \ {w} and Bg1= Bf1

\ {w}, Pg1= Pf1

. So Bg1∩ Bg =

(Bf1∩Bf ) \ {w} → Pf1 ∪ Pf ⊇ Pg1 ∪ Pg . Therefore g is a universal MTDF of G.

Conversely, suppose g is a universal MTDF of G. Extend g to be an MTDF f of G(v;w) by:

f(t) =

{0 if t = wg(t) otherwise.

We claim that f is universal for G(v;w). Let f1 be any MTDF of G(v;w). We must show thatBf1

∩Bf → Pf1∪ Pf . As before, we restrict f1 to be an MTDF g1 of G by:

g1(t) =

{f1(v) + f1(w) if t = vf1(t) otherwise.

Then Bg1 ∩Bg → Pg1 ∪Pg . Using the same technique as above and the fact that N(v) = N(w),it can be easily shown that Bf1

∩Bf → Pf1∪ Pf . Hence f is a universal MTDF of G(v;w).


The inverse of the splitting operation is to delete a vertex v fromG if there exists another vertexw such that N(v) = N(w). Let T (G) denote the class of graphs obtained by starting from G andsuccessively doing the splitting operation or inverse splitting operation. Then by Theorem 5, wehave the following corollary.

Corollary 2. G has a universal MTDF if and only if every graph in T (G) has a universal MTDF.

Further, from the proof of Theorem 5, we have the following corollary concerning 0–1 univer-sal MTDFs.

Corollary 3. G has a 0–1 universal MTDF if and only if every graph in T (G) has a 0–1 univer-sal MTDF.

Let Pn, Cn, and Kn denote a path with n nodes, a cycle with n nodes, and a complete graphwith n nodes, respectively. We also let Wn denote a wheel with n nodes (i.e., Wn = K1 +Cn−1)and F denote caterpillar graphs [3]. In [2, 3], it is proved that Pn, Cn,Kn,Wn and F all haveuniversal MTDFs.

So, we have the following corollary.

Corollary 4. T (Pn), T (Cn), T (Kn), T (Wn) and T (F ) all have universal MTDFs.

Proposition 5. The complete n-partite graph Km1,m2,...,mn has a universal MTDF.

Proof. The proposition follows from the above corollary and the fact that Km1,m2,...,mnis

in T (Kn).

4. CONCLUSION

We have given a stronger sufficiency condition than the one by Cockayne et al. [2] for an MTDFto be universal, and a splitting operation on graphs is introduced such that the universality ispreserved.

More research is needed to answer the following open problems.

(1) Theorem 3 is a necessary condition for an MTDF to be a universal MTDF. Is the conditionsufficient? The condition was shown to be necessary and sufficient when the given graphis a tree [1].

(2) Characterize graphs having universal MTDFs. Those graphs with a universal MTDF givea ‘‘yes’’ answer to the interpolation problem (see Section 1) raised by Hedetniemi.

(3) Is there a good algorithm for deciding whether a graph has a universal MTDF?

ACKNOWLEDGMENTS

The author would like to thank Joseph Cheriyan, Hugh Hind and Penny Haxell for helpful dis-cussions. The author also thanks the referees for their comments.

References

[1] E. J. Cockayne and C. M. Mynhardt, A characterization of universal minimal total dominating functionsin trees, Discrete Math., to appear.


[2] E. J. Cockayne, C. M. Mynhardt, and B. Yu, Universal minimal total dominating functions in graphs,Networks 24 (1994), 83–90.

[3] E. J. Cockayne, C. M. Mynhardt, and B. Yu, Total dominating functions in trees: minimality andconvexity, J. Graph Theory 19 (1) (1995), 83–92.

[4] G. Domke, S. T. Hedetniemi, and R. Laskar, Fractional packings, coverings, and irredundance ingraphs, Congr. Numer. 66 (1988), 227–238.

[5] S. T. Hedetniemi, private communication (1990).

[6] S. M. Hedetniemi, S. T. Hedetniemi, and T. Wimer, Linear time resource allocation algorithm for trees,technical Report, Department of Mathematical Sciences, Clemson University, URI-014 (1987).

[7] S. T. Hedetniemi and R. Laskar, Bibliography on domination in graphs and some basic definitions ofdomination parameters, Discrete Math. 86 (1990), 257–277.

[8] R. Laskar, A. Majumdar, G. Domke, and G. Fricke, A fractional view of graph theory, technical Report,Department of Mathematical Sciences, Clemson University, #576, June (1989).

[9] R. Laskar, J. Pfaff, S. M. Hedetniemi, and S. T. Hedetniemi, On the algorithmic complexity of totaldomination, SIAM J. Alg. Disc. Math. 5 (1984), 420–425.

[10] F. A. Valentine, Convex sets, McGraw-Hill, New York (1964).

[11] B. Yu, Convexity of minimal total dominating functions in graphs, Master's Thesis, University ofVictoria (1992).

Received June 22, 1994

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Convexity of minimal total dominating functions in graphs