DOI 10.1007/s10623-012-9792-1

**The exact values of the optimal average information ratio**

**of perfect secret-sharing schemes for tree-based access**

**structures**

**Hui-Chuan Lu** **· Hung-Lin Fu**

Received: 10 February 2012 / Revised: 15 December 2012 / Accepted: 18 December 2012 / Published online: 7 March 2013

© Springer Science+Business Media New York 2012

**Abstract** A perfect secret-sharing scheme is a method of distributing a secret among a
set of participants such that only qualified subsets of participants can recover the secret and
the joint shares of the participants in any unqualified subset is statistically independent of
the secret. The set of all qualified subsets is called the access structure of the scheme. In a
*graph-based access structure, each vertex of a graph G represents a participant and each edge*
*of G represents a minimal qualified subset. The information ratio of a perfect secret-sharing*
scheme is defined as the ratio between the maximum length of the share given to a participant
and the length of the secret. The average information ratio is the ratio between the average
length of the shares given to the participants and the length of the secret. The infimum of
the (average) information ratios of all possible perfect secret-sharing schemes realizing a
given access structure is called the (average) information ratio of the access structure. Very
few exact values of the (average) information ratio of infinite families of access structures
are known. Csirmaz and Tardos have found the information ratio of all trees. Based on their
method, we develop our approach to determining the exact values of the average information
ratio of access structures based on trees.

**Keywords** Secret-sharing scheme· Graph-based access structure · Average information
ratio· Entropy · Star covering · Tree

**Mathematics Subject Classification (2000)** 05C70· 94A60 · 94A62 · 94A17

Communicated by C. Blundo. H.-C. Lu (

### B

)Center for Basic Required Courses, National United University, Miaoli36003, Taiwan e-mail: hjlu@nuu.edu.tw; hht0936@seed.net.tw

H.-C. Lu· H.-L. Fu

**1 Introduction**

*A secret-sharing scheme is a method of distributing a secret among a set of participants in*
such a way that only qualified subsets of participants can recover the secret from the shares
they receive. If, in addition, the joint shares of the participants in any unqualified subset is
*statistically independent of the secret, then the secret-sharing scheme is called perfect. Since*
all secret-sharing schemes considered in this paper are perfect, we will simply use
*“secret-sharing scheme” for “perfect secret-“secret-sharing scheme”. The access structure of a secret-“secret-sharing*
*scheme is the collection of all qualified subsets in this scheme. It is required to be monotone*
which means any subset of participants containing a qualified subset must also be qualified.
There are two major tools for measuring the efficiency of a secret-sharing scheme, namely,
*the information ratio and the average information ratio. The information ratio of a *
secret-sharing scheme is the ratio between the maximum length (in bits) of the share given to a
participant and the length of the secret. The average information ratio of a secret-sharing
scheme is the ratio between the average length of the shares given to the participants and
the length of the secret. These ratios represent the maximum and average number of bits of
information the participants must remember for each bit of the secret. The lower the ratios
are, the lower storage and communication complexity the scheme has. Therefore, for a given
access structure, constructing a secret-sharing scheme with the lowest above-mentioned ratios
is one of the main goals of the research. The infimum of the (average) information ratios of
all possible secret-sharing schemes realizing a access structure is referred to as the (average)
information ratio of that access structure.

In 1979, Shamir [8] and Blakley [2] independently introduced the first kind of
secret-sharing schemes called the*(t, n)-threshold schemes in which the minimal qualified subsets*
*are the t-subsets of the set of participants of size n. Related problems have then received*
considerable attention. Secret-sharing schemes for various access structures and many
mod-ified versions of secret-sharing schemes with additional capacities were widely studied. The
reader is referred to [1,7] and their references for recent developments on secret-sharing
problems.

In the present paper, we only consider graph-based access structures. In such a structure,
*each vertex of a graph G represents a participant and each edge of G represents a minimal*
qualified subset. A secret-sharing scheme* for the access structure based on G is a collection*
of random variables*ξs*and*ξv*for*v ∈ V (G) with a joint distribution such that*

(i) *ξs*is the secret and*ξv*is the share of*v;*

*(ii) if uv ∈ E(G), then ξu*and*ξv*together determine the value of*ξs*; and

*(iii) if A⊆ V (G) is an independent set, then ξs*and the collection*{ξv|v ∈ A} are statistically*

independent.

*Given a discrete random variable X with possible values{x*1*, x*2*, . . . , xn*} and a probability

distribution*{p(xi)}ni*=1*, the Shannon entropy of X is defined as*
*H(X) = −*

*n*
*i*=1

*p(xi) log p(xi),*

*which is a measure of the average uncertainty associated with the random variable X. It*
*is well known that H(X) is a good approximation to the average number of bits needed*
*to represent the elements in X faithfully. Using Shannon entropy, the information ratio*
of the secret-sharing scheme* can be defined as R* = max*v∈V (G){H(ξv)/H(ξs)} and*

*the average information ratio as A R _{}* =

_{v∈V (G)}H(ξ_{v})/(|V (G)|H(ξs)). For*access structure based on G”. Also, “the information ratio (resp. the average *
*infor-mation ratio) of the access structure based on G” is referred to as “the inforinfor-mation*
*ratio (resp. the average information ratio) of G”, denoted as R(G) (resp. AR(G)). As*
*mentioned above, R(G) = inf{R| is a secret-sharing scheme on G} and AR(G) =*

inf*{AR _{}| is a secret-sharing scheme on G}. It is well known that R(G) ≥ AR(G) ≥ 1*

*and that R(G) = 1 iff AR(G) = 1. A secret-sharing scheme with R*

*= 1 or AR*= 1 is

*then called an ideal secret-sharing scheme. An access structure is ideal if there exists an ideal*
*secret-sharing scheme on it. Determining the exact value of R(G) or AR(G) is extremely*
challenging. It is not easy even for small graphs sometimes. Due to the difficulty, most known
*results give bounds on R(G) and AR(G). Stinson [*10] has shown the important bounds for
*general graphs: R(G) ≤* *d*+1

2 *where d is the maximum degree of G and A R(G) ≤* *2m2n+n*

*where n* *= |V (G)| and m = |E(G)|. The exact values of R(G) and AR(G) are obtained*
only for very few specific graphs. Most graphs of order no more than five, and the cycles and
paths have known exact values of the average information ratio [3,10]. Most graphs of order
no more than six, and the cycles, paths and trees have known exact values of the information
ratio [3,6,9–11*]. The information ratio of a tree T was determined by Csirmaz and Tardos*
[6*] as R(T ) = 2 −*1_{k}*where k is the maximum size of a core in T. Based on their method,*
*we develop our approach to the problem of determining the value of A R(T ) for any tree T.*
This paper is organized as follows. In Sect.2, some basic known results and definitions are
introduced. Our results are presented in Sects.3,4and5*. We derive a lower bound on A R(T )*
and introduce our approach in Sect.3. Our main results are shown in Sect.4. Subsequently,
in Sect.5*, two examples are given to demonstrate our systematic way of evaluating A R(T ).*

A concluding remark will be given in the final section.

**2 Preliminaries**

We introduce some basic known results on graph-based access structures first. The ideal
graph-based access structures have been completely characterized by Brickell and Davenport.
**Theorem 1 ([**4*]) Suppose that G is a connected graph. Then R(G) = 1 if and only if G is*

*a complete multipartite graph.*

*We introduce the methods of deriving upper bounds and lower bounds on A R(G) for a *
*non-ideal access structure G in what follows. By constructing a secret-sharing scheme on graph*
*G, one can obtain an upper bound ARon the average information ratio A R(G). Stinson’s*

decomposition construction [10] has been a major tool to do this job. This method enables
*us to build up secret-sharing schemes for graphs using complete multipartite coverings. A*
*complete multipartite covering of a graph G is a collection of complete multipartite subgraphs*
*Π = {G*1*, G*2*, . . . , Gl} of G such that each edge of G belongs to at least one subgraph in the*

collection. The value*l _{i}*

_{=1}

*|V (Gi)| is crucial for our discussion, we call it the vertex-number*

*sum of* *Π.*

**Theorem 2 ([**10*]) Suppose that{G*1*, G*2*, . . . , Gl} is a complete multipartite covering of a*

*graph G with V(G) = {1, 2, . . . , n}. Let Ri* *= |{ j|i ∈ V (Gj)}| and R = max*1*≤i≤nRi.*

*Then there exists a secret-sharing scheme on G with information ratio R _{}*

*and average*

*information ratio A R*

_{}where*R _{}= R and AR_{}* = 1

*n*

*n*

*i*=1

*Ri*= 1

*n*

*l*

*i*=1

*|V (Gi)|.*

*The only main tool for establishing lower bounds on A R(G) is the information theoretic*
approach [5]. Let* be a secret-sharing scheme in which ξs* is the random variable of the

secret and each*ξ _{v}* is the one of the share of

*v, v ∈ V (G). Define a real-valued function f*

*as f(A) = H({ξv*

*: v ∈ A})/H(ξs) for each subset A ⊆ V (G), where H is the Shannon*

*entropy. Then, A R _{}* = 1

_{n}

_{v∈V (G)}*f(v), where n = |V (G)|. Using properties of the entropy*

*function and the definition of a secret-sharing scheme, one can show that f satisfies the*following inequalities [5]:

*(a) f(∅) = 0, and f (A) ≥ 0;*

*(b) if A⊆ B ⊆ V (G), then f (A) ≤ f (B);*
*(c) f(A) + f (B) ≥ f (A ∩ B) + f (A ∪ B);*

*(d) if A⊆ B ⊆ V (G), A is an unqualified set and B is not, then f (A) + 1 ≤ f (B); and*
*(e) if neither A nor B is unqualified but A∩ B is, then f (A) + f (B) ≥ 1 + f (A ∩ B) +*

*f(A ∪ B).*

Csirmaz and Tardos [6*] defined a core V*0 *of a tree T as a subset V*0*of V(T ) such that*

*V*0*induces a connected subgraph of T and each vertex in V*0has a neighbor outside it. They

also showed the following theorem.

**Theorem 3 ([**6*]) Let V*0*be a core of a tree T. If f is defined as above, then*_{v∈V}_{0} *f(v) ≥*

2*|V*0*| − 1.*

*In the next section, we shall derive a lower bound on A R(T ) and rewrite Theorem*2as
*an upper bound on A R(T ) of particular form. Our approach can then be introduced.*
**3 Lower bound and upper bound on AR****(T)**

*Given a tree T, we let I N(T ) and L F(T ) be the sets of all internal vertices and leaves of*
*T respectively. Denote|I N(T )| as in(T ) and |L F(T )| as l f (T ). In order to cope with the*
*average information ratio, we extend the idea of a core of T. For T = K*1*,1, we define a*

*core cluster of T of size k as a partitionC* *= {V*1*, V*2*, . . . , Vk} of I N(T ) such that each*

*Vi, i ∈ {1, 2, . . . , k}, is a core of T. The size of a core clusterC* *is written as cC. We also*

*denote the minimum size of all core clusters of T as c*∗*(T ), called the core number of T.*
Note that*k _{i}*

_{=1}

*Vimay not be a core of T, if so, then c*∗

*(T ) = 1 for T = K*1

*,1. In addition,*

*we naturally define that c*∗*(K*1*,1) = 0.*

*The idea of a core cluster helps us establish a lower bound on A R(T ).*
**Theorem 4 If T*** = K*1*,1is a tree of order n, then AR(T ) ≥* *n+in(T )−c*

∗_{(T )}

*n* *.*

*Proof Suppose that is a secret-sharing scheme on T. Then the function f defined in*
Sect.2by the random variables from* satisfies all the properties (a) to (e) and Theorem*

3. Let*C* *= {V*1*, V*2*, . . . , Vk} be a core cluster of T. By Theorem*3and the definition of

a core cluster,_{v∈I N(T )}*f(v) =* *k _{i}*

_{=1}

_{v∈V}*i* *f(v) ≥*

*k*

*i*=1*(2|Vi| − 1) = 2in(T ) − k.*

*Since T is connected, f(v) ≥ 1 for all v ∈ V (T ) [*5]._{v∈V (T )}*f(v) =*_{v∈I N(T )}*f(v) +*

*v∈L F(T )* *f(v) ≥ 2in(T ) − k + l f (T ) = n + in(T ) − k. Thus we have AR* ≥ 1*n(n +*

*i n(T ) − k). Since the result holds for any secret-sharing scheme on T, we have AR(T ) ≥*

1

*n(n + in(T ) − c*∗*(T )).*

On the other hand, as suggested in Theorem2, in order to construct a secret-sharing scheme with lower average information ratio, we need a complete multipartite covering with the least vertex-number sum. Since we are dealing with trees, and stars are the only complete

multipartite trees, star coverings with the least vertex-number sum are what we are aiming
for. For a better description of our approach, given a star covering*Π of T with vertex-number*
*sum m, we define the deduction of Π, written d _{Π}, as d_{Π}*

*= |V (T )| + in(T ) − m. A star*covering with the largest deduction gives the least vertex-number sum. The largest value

*of the deductions over all star coverings of T is called the deduction of T and is denoted*

*as d*∗

*(T ). The following corollary is simply a rephrasement of Theorem*2in terms of the

*deduction of T.*

**Corollary 5 ([**10*]) LetΠ be a star covering of a tree T of order n, then*
*A R(T ) ≤n+ in(T ) − d*

∗_{(T )}

*n* *.*

Combining Theorem4and Corollary5, we have the following results.

**Theorem 6 For any star covering**Π of T and any core clusterCof T, c_{C}*≥ d _{Π}. In particular,*

*c*∗

*(T ) ≥ d*∗

*(T ).*

**Corollary 7 If there exists a star covering**Π of T and a core cluster*C* *of T such that*
*d _{Π}*

*= c*∗

_{C}, then d*(T ) = d*

_{Π}*= c*

_{C}*= c*∗

*(T ).*

*As indicated in these results, c*∗*(T ) = d*∗*(T ) makes a criterion for examining whether the*
*upper bound and the lower bound on A R(T ) will match. In the next section, we will show*
that this equality holds for all trees.

**4 The main results**

Blundo et al. [3*] gave an algorithm for producing a star covering of a tree T. We make a*
*slight modification to it and restate it for completeness. Let NT(v) be the set of all neighbors*

of*v in T and Sv*be the star centered at*v with NT(v) as its leaf set.*

Algorithm;

**Covering(T)** **Cover(v)**

Let*v ∈ I N(T )* *A(v) ← NT(v) ∩ I N(T )*

*Π ← φ* *Π ← Π ∪ {Sv*}

Cover*(v)* *E(T ) ← E(T )\E(S _{v})*

Output the star coveringΠ V (T ) ← V (T )\((N*T(v) ∩ L F(T )) ∪ {v})*

for all*v**∈ A(v) do Cover(v**)*

**Lemma 8 Let T be a tree. The star covering**Π of T produced by Covering(T ) has deduction*d _{Π}*

*= 1 if T = K*1

*,1and dΠ*

*= 0 if T = K*1

*,1.*

*Proof For T* * = K*1*,1, the initial vertex v and all leaves of T appear in exactly one star in Π.*

All internal vertices but the initial one appear twice in the covering. So the vertex-number
*sum m= l f (T ) + 1 + 2(in(T ) − 1) = |V (T )| + in(T ) − 1, and we have d _{Π}*

*= 1.*

We shall refine this process and obtain star coverings with higher deductions next.
A vertex*v ∈ I N(T ) is called a critical vertex of T if NT(v)∩L F(T ) = ∅. In the structure*

*of a tree T, critical vertices play an important role in our discussion. We use XT* to denote

*the set of all critical vertices of T. Let KTbe the subgraph induced by XT* and*T(resp. YT*)

*be the set of all nontrivial (resp. trivial) components in KT. The set YT* is in fact the set of all

*and E**⊆ E(T ), the graph T \V**is obtained by removing from T all vertices in V*as well
*as all edges incident to them. T\E**is resulted from removing all edges in E**from T. Both*
*T\V**and T\E*may contain isolated vertices.

**Proposition 9 Let T*** = K*1*,1be a tree. IfT* *= ∅ and |YT| = y ≥ 0, then there exists a*

*star coveringΠ of T with deduction dΠ* *= y + 1.*

*Proof Let G be an arbitrary component in T\YT. If w*1*, . . . , wlare all of the vertices in YT*

*that are adjacent to some vertices in G, then we define ˜G as the subgraph of T induced by*
*V(G) ∪ {w*1*, . . . , wl}. Let H = { ˜G|G is a component in T \YT} and Π˜G*be the star covering

produced by algorithm Covering( ˜G). By the definition of Y*T, no ˜G is isomorphic to K*1*,1,*

*so d _{Π}_{˜G}* = 1 by Lemma8. Since

_{˜G∈H}E( ˜G) = E(T ), the covering Π =*is a*

_{˜G∈H}Π_{˜G}*star covering of T with vertex-number sum*

*m* =
*˜G∈H*
*(|V ( ˜G)| + in( ˜G) − 1)*
=
⎛
*⎝V (T ) +*
*v∈YT*
*(degT(v) − 1)*
⎞
*⎠ + (in(T ) − y)*
−
⎛
⎝
*v∈YT*
deg* _{T}(v) − (y − 1)*
⎞
⎠

*= V (T ) + in(T ) − (y + 1).*

*Next, we consider the core number of T. For a tree T with XT* *= ∅, {I N(T )} is obviously*

a core cluster of minimum size. The following lemma is straight forward.
**Lemma 10 Let T*** = K*1*,1be a tree. If XT* *= ∅, then c*∗*(T ) = 1.*

Now, we introduce the way we decompose a tree in order to define a core cluster we
*need. Let V* *⊆ V (T ). Given a vertex ¯v ∈ NT(v) ∩ I N(T ) for each v ∈ V**, we set*

*E* *= {v ¯v|v ∈ V**}. For each component G in T \E**, let G*+ *be the subtree of T obtained*
*by attaching to G all edges of the formv ¯v if ¯v ∈ V (G), then G*+ *= G if G does not*
contain any*¯v. We also denote the collection of all G*+*’s, where G is a component in T\E**,*
asH+*(T, V**, E**). Observe that, if deg _{T}(v) = 2, then v ∈ L F(G*+

*) for exactly two G*+’s in the collectionH+

*(T, V*

*, E*

*).*

**Proposition 11 Let T*** = K*1*,1* *be a tree. IfT* *= ∅ and |YT| = y ≥ 0, then c*∗*(T ) =*

*d*∗*(T ) = y + 1.*

*Proof It suffices to show that there is a core cluster of T of size y+ 1. For each v ∈ YT,*

choose an arbitrary neighbor of*v as ¯v, then ¯v ∈ I N(T ). Let E**= {v ¯v|v ∈ YT}. There are*

*y*+ 1 subgraphs in H+*(T, YT, E**). Let H*+*(T, YT, E**) = {G*+0*, G*+1*, . . . , G*+*y} where Gi*’s,

*i= 0, 1, . . . , y are the components in T \E**. Note that any two vertices in YT*have distance

*at least two, so I N(G*+_{i}*) = ∅. Let Vi* *= I N(G*+_{i}*) ∪ {v|v ∈ V (Gi) ∩ YT* and deg*T(v) = 2}.*

We claim that*{V*0*, V*1*, . . . , Vy} is a core cluster of T. First, each vertex u ∈ I N(T )\YT*

*belongs to exactly one I N(G*+_{i}*) and also exactly one Vi. Each v ∈ YT* belongs to exactly

*It belongs to exactly one I N(G*+_{i}*) and hence exactly one Vi. If degT(v) = 2, then v is a*

*leaf of exactly one component Giin T\E*and is a leaf of two subgraphs inH+*(T, YT, E**).*

*Hence it belongs to exactly one Vi* *and none of I N(G*+*j)’s, j = 0, 1, . . . , y. This shows*

that*{V*0*, V*1*, . . . , Vy} is a partition of I N(T ). Next, each Vi* certainly induces a connected

*subgraph of T. In addition, each v ∈ Vi∩ YT* has a neighbor *¯v not in Vi. Each u ∈ Vi\YT*

*has a leaf neighbor in T which does not belongs to Vi. Hence, Vi* *is a core of T. Since we*

*have a core cluster of size y+ 1, the result then follows immediately by Proposition*9and

Corollary7.

Before literally proving our main theorem, we examine the relation between the deductions
of star coverings of subtrees inH+*(T, V**, E**) and the deduction of a star covering of T more*
closely.

**Lemma 12 Let V***be an independent subset of I N(T ) and z = |{v ∈ V*|
deg_{T}(v) ≥ 3}|. For each v ∈ V*, let ¯v be a nonleaf neighbor of v in T and E**= {v ¯v|v ∈ V**}.*
*If there is a star coveringΠT**of each T*∈ H+*(T, V**, E**) with deduction dΠT , then Π =*

*T*∈H+*(T,V**,E**)ΠT**is a star covering of T with deduction dΠ* =

*T*∈H+*(T,V**,E**)dΠT −z.*

*Proof Denote*H+*(T, V**, E**) by H*+for now. Since_{T}_{∈H}+*E(T**) = E(T ), Π is a star*
*covering of T. The vertex-number sum m of Π is*

*m* =
*T*∈H+
*(|V (T*_{)| + in(T}_{) − d}*ΠT )*
*= |V (T )| + |V*_{| + in(T ) − (|V}_{| − z) −}*T*∈H+
*d _{Π}_{T }*

*= |V (T )| + in(T ) −*⎛ ⎝

*T*∈H+

*d*

_{Π}_{T }*− z*⎞

*⎠ .*

Now, we are in a position to present our main theorem.
* Theorem 13 Let T be a tree of order n, then c*∗

*∗*

_{(T ) = d}

_{(T ) and}*A R(T ) =* *n+ in(T ) − c*∗*(T )*

*n* *.*

*Proof We prove this result by induction on|XT|.*

(1) If*|XT| = 0 or 1, then T* *= ∅. The result holds by Proposition*11.

(2) Suppose that*|XT| ≥ 2. By Proposition*11, we may assume that*T* * = ∅.*

Choose a vertex*v ∈ L F(T**) for some T**∈ T* and let *¯v be the neighbor of v in T**. There*

*are two subtrees G*+_{0} *and G*+_{1} inH+*(T, {v}, {v ¯v}), each of which is not a K*1*,1. Let G*+0 be

the one not containing *¯v, then |X _{G}*+

0*| < |XT| is obviously true. Since v ∈ L F(G*
+

1*), it is*

*no longer a critical vertex of G*+_{1}*, we also have |X _{G}*+

1*| < |XT|. By induction hypothesis,*

there exist a star covering *Πi* *of G*+*i* and a core cluster *Ci* *= {Vi 1, Vi 2, . . . , Vi ki*} with

*dΠi* *= cCi* *= ki* *> 0, i = 0, 1. Then Π = Π*0*∪ Π*1*is a star covering of T. We construct a*

*core cluster of size d _{Π}* next.

(i) If deg* _{T}(v) ≥ 3, then d_{Π}= k*0

*+ k*1− 1 by Lemma12. Suppose that

*v ∈ V*01

*. Since V*01

*of G*+_{0} because*v is critical both in T and in G*+_{0}*. We may assume that v**∈ V*02*. Now, let*

*C= {V*01*∪ V*02*, V*03*, . . . , V0k*0*, V*11*, . . . , V1k*1*}, then |C| = k*0*+ k*1*− 1. We claim thatC*

*is a core cluster of T. First note that I N(G*+_{0}*) ∪ I N(G*+_{1}*) = I N(T ) and any two sets in*

*C*are disjoint. Each set in*C\{V*01*∪ V*02*} is a core of G*+_{0} *or G*+_{1}*, hence a core of T. For*

*V*01*∪ V*02*, ¯v is a neighbor of v in T not in V*01*∪ V*02*. Since v ∈ L F(T**), v*is not critical

and then has a leaf neighbor*v** = v in G*+_{0} *(and in T ) not in V*02*, so v* */∈ V*01*∪ V*02and

*V*01*∪ V*02*is qualified as a core of T. Therefore,Cis a core cluster of T of size dΠ.*

(ii) If deg_{T}(v) = 2, then d_{Π}*= k*0*+ k*1by Lemma12. Since*v is a critical vertex of T, the*

neighbor*v* * = ¯v in T is an internal vertex of G*+_{0}*. We may assume that v* *∈ V*01*. Let*

*C* *= {V*01*∪ {v}, V*02*, . . . , V0k*0*, V*11*, . . . , V1k*1*}, then |C| = k*0*+ k*1*. To show thatC*is

*a core cluster of T, it suffices to show that V*01*∪ {v} is a core of T. Note that v*is not

*critical in both G*+_{0} *and T. It has a leaf neighbor v** = v not in V*01*∪ {v}. Besides, ¯v is a*

neighbor of*v in T not in V*01*∪ {v}. V*01*∪ {v} is then a core of T. Therefore, T also has*

*a core cluster of size dΠ*in this case.

*In both cases, we have c*∗*(T ) = d*∗*(T ), which implies that the lower bound and the upper*
*bound of A R(T ) coincide. Hence, AR(T ) =* *n+in(T )−c*∗*(T )*

*n* *.*

**5 Some examples**

In this section, we evaluate the average information ratio systematically for two infinite classes of trees using our approach.

The only infinite class of trees which has known average information ratio is the paths. By evaluating the core number, we can easily obtain the known result.

**Proposition 14 ([**10*]) Let Pnbe a path of length n. Then*

*A R(Pn) =*
*3n*
2*(n+1), i f n is even; and*
*3n*+1
2*(n+1), i f n is odd.*

*Proof By Proposition*11*, we have c*∗*(P*1*) = 0, c*∗*(P*2*) = c*∗*(P*3*) = 1 and c*∗*(P*4*) = 2.*

Observe that*Pn* *= {Pn*−4*} for all n ≥ 5. Since any leaf of the Pn*−4in*Pn* has degree two

*in Pn, from the proof of Theorem*13*, we have c*∗*(Pn) = c*∗*(Pn*−4*) + 2. Recursively, we have*

*c*∗*(Pn) =*
*c*∗*(Pi) + 2k,* *if n= 4k + i, i = 1, 2, 3; and*
*c*∗*(P*4*) + 2(k − 1), if n = 4k.*
=
*n*
2*,* *if n is even; and*
*n*−1
2 *, if n is odd.*
Hence,
*A R(Pn) =* *(n + 1) + (n − 1) − c*
∗_{(P}*n)*
*n*+ 1 =
*3n*
2*(n+1), if n is even; and*
*3n*+1
2*(n+1), if n is odd.*

*Next, we evaluate the average information ratio of complete q-ary trees. A complete q-ary*
*tree with k levels is a rooted tree such that each nonleaf vertex has q children and the distance*
*from the root to each leaf is k.*

**Proposition 15 Let T**kbe a complete q-ary tree with k levels, q≥ 2. Then*A R(Tk) =*
⎧
⎨
⎩
*qk*+2*+2qk*+1*−q*2*−2q*
*(q+1)(qk*+1_{−1)}*,* *i f k i s even; and*
*qk*+2* _{+2q}k*+1

*2*

_{−q}

_{−q−1}*(q+1)(qk*+1

_{−1)}*, i f k is odd.*

*Proof By Proposition*11*, c*∗*(T*1*) = 1 and c*∗*(T*2*) = 2. Observe that Tk* *= {Tk*−2} and

*the Tk*−2*has qk*−2 *leaves, each of which has degree q+ 1 ≥ 3 in Tk. Since each leaf of*

*the Tk*−2*and its descendants in Tkcompose a T*2*, from the proof of Theorem*13, we get

*c*∗*(Tk) = c*∗*(Tk*−2*) + qk*−2*(c*∗*(T*2*) − 1) = c*∗*(Tk*−2*) + qk*−2*. Recursively, the core number*

*of Tk*can be evaluated as follows.

*c*∗*(Tk) =*
*qk*−2* _{+ q}k*−4

*2*

_{+ · · · + q}*∗*

_{+ c}*2*

_{(T}*), if k is even; and*

*qk*−2

*+ qk*−4

*+ · · · + q + c*∗

*(T*1

*), if k is odd.*= ⎧ ⎨ ⎩

*qk+q*2−2

*q*2

_{−1}

*,*

*if k is even; and*

*qk*2

_{+q}

_{−q−1}*q*2

_{−1}

*, if k is odd.*Therefore,

*A R(Tk) =*

*qk*+1

_{−1}

*q*−1 +

*qk*

_{−1}

*q*−1

*− c*∗

*(Tk)*

*qk*+1

_{−1}

*q*−1 = ⎧ ⎨ ⎩

*qk*+2

*+2qk*+1

*−q*2

*−2q*

*(q+1)(qk*+1

_{−1)}*,*

*if k is even; and*

*qk*+2

*+1*

_{+2q}k*2*

_{−q}

_{−q−1}*(q+1)(qk*+1

_{−1)}*, if k is odd.*

**6 Conclusion**

*We have proposed the idea of the deduction d*∗*(T ) and the core number c*∗*(T ) of a tree and*
showed that these values are the same, thereby proving the upper bound and the lower bound
of the average information ratio of a tree coincide. By doing so, we also present a systematic
way of evaluating the core number of a tree. Together with the result by Csirmaz and Tardos
[6], we complete the work of evaluating the information ratio and the average information
ratio of all trees.

In fact, the notions of the deduction and the core number can be extended to general graphs.
*The condition d*∗*(G) = c*∗*(G) makes a criterion for examining whether the upper bound*
*and the lower bound on A R(G), for any G, will match. The idea formulates a complicated*
problem of secret-sharing schemes into a problem in graph theory with easy description.
“For what kind of graphs will the identity be true?” is indeed an interesting question to
*investigate. One obvious restriction to set on G is that G must be of larger girth. A star*
covering generally does not serve as a complete multipartite covering with the least
vertex-number sum for a graph of small girth. We have made some progress in the study of bipartite
graphs of large girth. Finding a star covering whose deduction matches the size of a core
cluster is in general very difficult. However, there have not been any bounds or asymptotic
results on the complexity of the problem yet.

**Acknowledgments** The authors would like to express their deep gratefulness to the reviewers for their detail
comments and valuable suggestions which lead to great improvement in the presentation of the paper. The
work of Hui-Chuan Lu was supported in part by NSC 100-2115-M-239-001 and the work of Hung-Lin Fu
was supported in part by NSC 97-2115-M-009-011-MY3.

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