Liar’s Domination

Peter J. SlaterDepartment of Mathematical Sciences and Computer Sciences Department, The University of Alabama inHuntsville, Huntsville, Alabama 35899

Assume that each vertex of a graph G is the possible loca-tion for an “intruder” such as a thief, or a saboteur, a fire ina facility or some possible processor fault in a computernetwork. A device at a vertex v is assumed to be ableto detect the intruder at any vertex in its closed neigh-borhood N [v ] and to identify at which vertex in N [v ] theintruder is located. One must then have a dominating setS ⊆ V (G), a set with ∪v∈SN [v ] = V (G), to be able to iden-tify any intruder’s location. If any one device can fail todetect the intruder, then one needs a double-dominatingset. This article introduces the study of liar’s dominatingsets, sets that can identify an intruder’s location evenwhen any one device in the neighborhood of the intrudervertex can lie, that is, any one device in the neighbor-hood of the intruder vertex can misidentify any vertex inits closed neighborhood as the intruder location. Liar’sdominating sets lie between double-dominating sets andtriple-dominating sets because every triple-dominatingset is a liar’s dominating set, and every liar’s dominat-ing set must double dominate. © 2009 Wiley Periodicals, Inc.NETWORKS, Vol. 54(2), 70–74 2009

Keywords: domination; detection; location; fault-tolerant

1. INTRODUCTION

A graph G = (V , E) might be used to model a facilitywith each vertex in V(G) representing an area of the facilitysuch as a room, hallway, or ventilation duct. Likewise, graphG might represent a computer network where each vertex v ∈V(G) represents a processor. Edges of G could link verticesrepresenting adjacent areas of the facility or processors withdirect links. A facility area or a processor will be identifiedwith the vertex that represents it.

Facilities are subject to having an “intruder” such as a thief,saboteur, or fire that must be detected and have its locationprecisely identified. Similarly, in a multiprocessor systema faulty processor (herein also called an intruder) must belocated. In general, it is assumed here that the possible loca-tions for the single intruder are all of the vertices in V(G).

Received April 2007; accepted July 2008Correspondence to: P. J. Slater; e-mail: slater@math.uah.eduDOI 10.1002/net.20295Published online 13 February 2009 in Wiley InterScience (www.interscience.wiley.com).© 2009 Wiley Periodicals, Inc.

The neighborhood of a vertex v ∈ V(G) is the set of verticesadjacent to it, N(v) = {x ∈ V(G) : vx ∈ E(G)}, and theclosed neighborhood is N[v] = N(v)∪{v}. It is assumed thata protection device placed at a vertex v can detect the pres-ence of an intruder precisely when the intruder is in N[v].When a protection device at vertex v can distinguish betweenthere being an intruder at v or at a vertex in N(v), but whichvertex in N(v) cannot be pinpointed, then one is interested inhaving a locating-dominating set. Locating sets were intro-duced in Slater [41] and subsequently by Harary and Melter[19] where they were called metric bases. The concepts oflocating and dominating were merged in [42, 43]. Furtherstudies of locating-dominating sets include [6, 11, 13–15,17, 21, 23, 28, 29, 37, 44, 45]. When only the presence ofan intruder in N[v] can be detected, with no information asto which vertex in N[v] contains the intruder, one is inter-ested in identifying-codes as in [1–5, 7–10, 12, 16, 22, 24–27,30–36, 38, 40]. Fault-tolerant locating-dominating sets areconsidered in [45].

In this article a protection device at vertex v is assumedto be able to detect an intruder at any vertex in N[v] and,moreover, to be able to precisely identify at which vertex inN[v] the intruder is located. Thus, a set D ⊆ V(G) chosen tobe the set of locations for protection devices will be able tolocate any intruder location if and only if D is a dominatingset, that is, ∪w∈DN[w] = V(G). The minimum cardinality ofa dominating set for G is denoted by γ (G). (See [20].)

Each protection device actually has two functions. Thefirst is to correctly detect the presence of an intruder, and thesecond is to accurately report the correct location. Assumewe desire to have some fault-tolerance. Specifically, assumewe must allow for any one protection device to be faulty. If adevice might fail to detect the presence of an intruder that isactually in its closed neighborhood, then the set D of locationsfor the protection devices will be a single-fault-tolerant set ifand only if it is a double-dominating set, that is, |N[x]∩D| ≥ 2for every x ∈ V(G). As defined by Harary and Haynes [18],vertex set D ⊆ V(G) is a k-tuple dominating set if |N[x] ∩D| ≥ k for every x ∈ V(G), and the minimum cardinality ofa k-tuple dominating set for G is denoted by γxk(G). A 2-tuple dominating set is also called a double-dominating set,and γx2(G) is also denoted by dd(G). Obviously, a 3-tupledominating set is also called a triple-dominating set.

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This article introduces the study of liar’s dominating setsfor which it is assumed that any one protection device in theneighborhood of the intruder vertex might (either deliberatelyor through a transmission error) misreport the location of anintruder in its closed neighborhood. That is, what is soughtis a single-fault-tolerant protection-placement set where adevice’s fault could be in identifying the location.

2. LIAR’S DOMINATING SETS

A dominating set D ⊆ V(G) is a liar’s dominating setif for any designated vertex x ∈ V(G) (namely, the intruderlocation) if all or all but one of the vertices in N[x]∩D reportvertex x as the intruder location, and at most one vertex w inN[x] ∩ D either reports a vertex y ∈ N[w] or fails to reportany vertex, then the vertex x can be correctly identified as thedesignated vertex. That is, if an intruder is at any vertex xthen the protection devices outside N[x] are assumed to notreport any intruder (that is, we do not allow a “false alarm”),one vertex w ∈ N[x] ∩ D can report nothing or any vertex inN[w] as the intruder location, every other element of N[x]∩Dwill correctly report location x, and x will be identifiable.Note that every superset of a liar’s dominating set is also aliar’s dominating set. The minimum cardinality of a liar’sdominating set for graph G, the liar’s domination numberof G, will be denoted by γLR(G). Note that it is assumed thatall protection devices can detect correctly and there will beat most a single fault in the reporting.

In general, for any vertex set L ⊆ V(G), we will say thatvertex x is LR-dominated by L if L can correctly identify xas a designated vertex.

Theorem 1. If L ⊆ V(G) is a liar’s dominating setthen each component of 〈L〉, the subgraph generated by L,contains at least three vertices.

Proof. If 〈L〉 has a singleton component {w}, then adevice at location w can be the faulty one and fail to reportan intruder at w itself. That is, L does not LR-dominate w. If〈L〉 has a 2-element component {v, w}, suppose that device vreports w and device w reports v (or else that each of v and wreports itself), then, by symmetry, the intruder can be at v orat w. That is, L does not LR-dominate v or w. ■

Henceforth, it is assumed that each component of G hasorder at least three.

Observation 2.

a. If two vertices in vertex set L report x to be the intruderlocation, then at least one of them is not misreporting, so xmust actually be the intruder location.

b. If two protectors u and v in L contradict each other (forexample, w ∈ N[u] ∩ N[v] and u reports an intruder at wwhile v does not), then every x ∈ L − {u, v} is reportingcorrectly because u or v is the (single) liar.

Theorem 3. For every connected graph G of order n ≥ 3we have γx2(G) ≤ γLR(G), and, if G has minimum degreeδ(G) ≥ 2, then γx2(G) ≤ γLR(G) ≤ γx3(G).

FIG. 1. γLR(G1) = 4.

Proof. Let L be a γLR(G)-set. Any x ∈ V(G) with|N[x] ∩ L| ≤ 1 would not be LR-dominated by L, so L mustbe a double-dominating set. Note that G has a 3-tuple domi-nating set if and only if δ(G) ≥ 2. Let S be a γx3(G)-set. Forany designated vertex x we have |N[x] ∩ S| ≥ 3 and at mostone device location in S would be faulty. So, by Observation2a, S is a liar’s dominating set. ■

Consider a possible liar’s dominating set L for the graphG1 in Figure 1. Because L is a double-dominating set wehave N[v3] = {v2, v3} ⊆ L, and v1 is double dominated soL ∩ {v1, v4} = . Consider L1 = {v1, v2, v3}. Assume v1

reports itself as an intruder location, v2 reports v4, and v3

does not report. The intruder could be at v1 with v2 being theliar, or the intruder could be at v4 with v1 being the liar. Itfollows that γLR(G1) = 4.

Theorem 4. Vertex set L ⊆ V(G) is a liar’s dominating setif and only if (1) L double dominates every v ∈ V(G) and(2) for every pair u,v of distinct vertices we have |(N[u] ∪N[v]) ∩ L| ≥ 3.

Proof. Assume L is a liar’s dominating set for G. Asnoted in the proof of Theorem 3, L must be a double-dominating set. Also, because L is double dominating, if|(N[u] ∪ N[v])∩ L| < 3 then we must have |(N[u] ∪ N[v])∩L| = 2. Suppose that (N[u] ∪ N[v]) ∩ L = {x, y} (where{x, y} ∩ {u, v} = is possible). If protection device x reportslocation u and protection device y reports location v, then theintruder location could be u with y the liar or it could be vwith x the liar. So, L would not be a liar’s dominating set.

For the converse, assume conditions (1) and (2) hold. Note,in particular, that these conditions imply that each componentof 〈L〉 has order at least three. Note also that condition (1)implies that whenever there is an intruder at least one elementin L reports the correct location. Also, by assumption there isat most one liar, and it follows that at most two vertices canbe identified as the possible intruder location. Assume firstthat only one vertex x is being reported. Because of (1), x isdominated by at least two elements of L and at least one ofthem is telling the truth, so x must be the intruder location.Assume second that two vertices are being reported, say a ∈ Lreports intruder location u ∈ N[a] and b ∈ L reports intruderlocation v ∈ N[b]. One of a or b is the liar and by (2) wehave a third vertex w ∈ (N[u] ∪ N[v]) ∩ L. Then w correctlyidentifies u or v as the intruder location. ■

Theorem 5. If graph G of order n = |V(G)| has maximumdegree �(G) = r (in particular, if G is regular of degree r),then γLR(G) ≥ (6/(3r + 2))n.

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FIG. 2. γLR(T31) = 24 = (3/4)(n + 1).

Proof. Let L ⊆ V(G) be a γLR(G)-set with γLR(G) = k.Then

∑v∈L deg(v) ≤ kr. Note that an edge in 〈L〉 contributes

two to∑

v∈L deg(v). By Theorem 1, the number of edgesin the subgraph 〈L〉 satisfies |E(〈L〉)| ≥ (2/3)k. Thus thenumber of edges between L and V(G) − L is at most kr −(4/3)k, and each v ∈ V(G) − L has at least two neighborsin L. Consequently, |V(G) − L| ≤ (1/2)k(r − 4/3). Thusn ≤ k + (1/2)k(r −4/3) = k(r/2+1/3), and k = γLR(G) ≥(6/(3r + 2))n. ■

Corollary 6. For a cycle Cn we have γLR(Cn) = �(3/4)n�.

Proof. Cycle Cn is regular of degree 2, so γLR(G) ≥�(6/8)n�. To see that γLR(Cn) ≤ �(3/4)n�, let V(Cn) ={v1, v2, . . . , vn} and E(Cn) = {vivi+1 (mod n) : 1 ≤ i ≤ n}.Then L = V(Cn) − {v4, v8, v12, . . .} is an LR-dominatingset. ■

Theorem 7. For graph G of order n = |V(G)| and sizem = |E(G)|, we have γLR(G) ≥ (3/4)(2n − m).

Proof. Let L be a γLR-set. By Theorems 1 and 4,|E(〈L〉)| ≥ (2/3)|L| and each vertex in V(G) − L has atleast two neighbors in L. Consequently, m ≥ 2(n − |L|) +(2/3)|L| = 2n − (4/3)|L| = 2n − (4/3)γLR(G), whichimplies that γLR(G) ≥ (3/4)(2n − m). ■

In particular, a γLR(T)-set for a tree T of order n mustcontain more than (3/4)n vertices.

Corollary 8. For a tree T of order n ≥ 3, (3/4)(n + 1) ≤γLR(T) ≤ n.

For the lower bound of Corollary 8, tree T31 in Figure 2illustrates how to obtain an infinite family F of trees Twith γLR(T) = (3/4)(|V(T)| + 1). Namely, let P3 ∈F with V(P3) = {v1, v2, v3} and E(P3) = {v1v2, v2v3}.Given any T ∈ F where V(T) = {v1, v2, . . . , v4k−1},let T∗ be the tree obtained from T by adding verticesv4k , v4k+1, v4k+2, and v4k+3, and adding edges v4k+1v4k+2

and v4k+2v4k+3, adding exactly one of the two edgesv4kv4k+1 and v4kv4k+2, and adding a final edge v4kvi where

i∈{1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, . . . , v4k−7, v4k−6, v4k−5,v4k−3, v4k−2, v4k−1}. Let T∗ ∈ F . Note that |V(T∗)| =4k + 3 and γLR(T∗) = 3k + 3 where L = V(T∗) −{v4, v8, v12, . . . , v4k} is a γLR(T∗)-set. It is shown in Rodenand Slater [39] that F is precisely the set of all trees T withγLR(T) = (3/4)(|V(T)| + 1).

Proposition 9. For a path Pn of order n, γLR(Pn) =�(3/4)(n + 1)�, and γLR(Pn) = (3/4)(n + 1) if and onlyif n = 4k + 3.

On the other hand, lots of trees T have γLR(T) = |V(T)|.For example, by Theorem 4 if every vertex v ∈ V(T) isan endpoint or a support vertex (that is, v is adjacent to anendpoint w), then V(T) is the only γLR(T)-set. The family oftrees T with γLR(T) = |V(T)| is actually more general thanthis.

Theorem 10. For a tree T of order n, γLR(Tn) = n ifand only if each v ∈ V(T) is an endpoint or at least onecomponent of T − v has cardinality at most two.

Proof. Every γLR(T)-set is double dominating, so it con-tains each endpoint (and each support vertex). Theorem 1shows that any vertex v for which T − v has a component ofcardinality at most two must be in every γLR(T)-set. Hencethe condition is sufficient to guarantee that γLR(T) = n.

For the converse, assume v is not an endpoint and eachcomponent of T − v has at least three vertices. Conditions(1) and (2) of Theorem 4 apply to L = V(T) − v, and henceγLR(T) ≤ n − 1. ■

One can use Theorem 10 to easily see that there are exactlythree trees of order n ≤ 7 with γLR(T) < |V(T)|. These threeare shown in Figure 3.

3. COMPLEXITY

Determining the liar’s domination number of a graph canbe cast as a decision problem as follows:

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FIG. 3. Trees T of order seven with γLR(T) = 6.

LIAR’S DOMINATING SET (LR-DOM)Instance: Graph H = (V(H), E(H)), positive integer J ≤

|V(H)|Question: Does there exist a liar’s dominating set L for H such

that |L| ≤ J? (That is, is γLR(H) ≤ J?)

LR-DOM can be shown to be NP-complete by reducingthe known NP-complete dominating set problem to it.

DOMINATING SET (DOM)Instance: Graph G = (V(G), E(G)), positive integer K ≤

|V(G)|Question: Does there exist a dominating set D for G such that

|D| ≤ K? (That is, is γ (G) ≤ K?)

Theorem 11. LR-DOM is NP-complete.

Proof. Using Theorem 4 one can easily see that LR-DOM is in NP. Given an instance of DOM, graph G =(V(G), E(G)) where V(G) = {v1, v2, . . . , vn} and K ∈ Z+,K ≤ |V(G)|, construct graph H as follows: V(H) = V(G) ∪{xi, yi, zi : 1 ≤ i ≤ n} and E(H) = E(G) ∪ {vixi, xiyi, yizi :1 ≤ i ≤ n}. Let J = K + 3n. We will show that γ (G) ≤ K ifand only if γLR(H) ≤ J .

First, assume D ⊆ V(G) is a γ (G)-set with |D| ≤ K . Letvi ∈ L in H if and only if vi ∈ D in G. Also let {xi, yi, zi} ⊆ Lfor 1 ≤ i ≤ n. Thus |L| = |D| + 3n ≤ K + 3n. Clearly eachof xi, yi, zi for 1 ≤ i ≤ n is double dominated by L. Everyvi is dominated once by xi and at least once by some x ∈ D.Therefore every vi, 1 ≤ i ≤ n, is double dominated. Henceevery vertex in V(H) is double dominated by L. Now considerevery pair of vertices in V(H). Because vi is dominated by xi

and some x ∈ D then |(N[vi] ∪ N[vk]) ∩ L| ≥ |{xi, xk , x}| =3, |(N[vi] ∪ N[xk]) ∩ L| ≥ |{xi, x, yk}| = 3, and |(N[vi] ∪N[zk])∩L| ≥ |{xi, x, yk , zk}| = 4. Also |(N[xi]∪N[zk])∩L| ≥|{xi, yi, zk}| = 3. Notice that if some yi is in the pair thenthe pair is triple dominated by {xi, yi, zi}. Thus, every pairof vertices in V(H) is triple dominated. Thus by Theorem4, L is a liar’s dominating set. So γLR(H) ≤ |L| ≤ K+3n = J .

Second, assume L ⊆ V(H) is a γLR(H)-set with |L| ≤K + 3n. Because every endpoint and support vertex mustbe in L and every component of 〈L〉 must contain at leastthree vertices, then {xi, yi, zi} ⊆ L, 1 ≤ i ≤ n. These accountfor 3n of the vertices in L. Since each vi, 1 ≤ i ≤ n, isdouble dominated by L, once by xi, then each vi must also bedominated by some x ∈ V(G)∩L. Thus V(G)∩L dominatesV(G) and |V(G)∩L| ≤ K . Therefore γ (〈{v1, v2, . . . , vn}〉) =γ (G) ≤ K . ■

4. OBSERVATIONS

Further results on this topic will appear in Roden andSlater [39]. We also have under study cases involving multipleintruders or multiple faults (liars).

Theorem 1 suggests some interesting questions even whennot considering fault-tolerance. We can let ijxk(G) (respec-

tively, Ijxk(G)) be the minimum cardinality of a k-tuple

dominating set D ⊆ V(G) where each component of 〈D〉has cardinality at most j (respectively, cardinality at least j).The domination number γ (G) = I1

x1(G) and, more generally,the k-tuple domination number of G is γxk(G) = I1

xk(G).The independent domination number is i(G) = i1x1(G), andthe total domination number is γt(G) = I2

x1(G). (Note thatγxk(G) = I1

xk(G) = I2xk(G) = · · · = Ik

xk(G).)We are studying the general parameters suggested by

Theorem 4, namely, γx(c1,c2,...,ct)(G) which is the minimumcardinality of a set L ⊆ V(G) such that |(∪v∈SN[v])∩L| ≥ ci

for all S ⊆ V(G) with |S| = i, for i ≤ i ≤ t. Note thatTheorem 4 states that γLR(G) = γx(2,3)(G).

Finally, we are beginning to study liar’s identifying codesand liar’s locating-dominating sets.

Acknowledgments

This paper is dedicated in memory of Frank Boesch, ahard worker and a good man.

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