Our second construction is similar to the first, while it performs better than Construction I when k ≥ 31. The major difference is that Bl is replaced
with Gl in the covering. With the notations used before, we define our
2⌋ is the complete multipartite covering of the ⌊k2⌋-weighted subgraph W¯ = W that the edges not in the subgraphs W
a1, . . . , a⌊k−1 Lemma 2.4.1. The collection eΠk is a complete multipartite covering of Gk
with minimum edge occurrence one.
Figure 2.2: The binary tree for Construction (II)
In order to evaluate the vertex-number sum emk of eΠk, we consider the reduced form first. Let eΠ0k and em0k be the reduced version of eΠk and emk
and Mj,2j0 to compose G02j or go with G0j and Mj+1,2j+10 to compose G02j+1. Recursively, all G0k’s can be obtained by using this process repeatly from G1, G2, G3 and some Mi,k0 ’s. As we have done in Section 2.3, this relation is depicted by a binary tree in Figure 2.2. The 2xpaths of length x from the root give the conformations of the 2x k-weight graphs where 2x+1 ≤ k ≤ 3 · 2x− 1 or 3 · 2x ≤ k ≤ 2x+2 − 1.
Theorem 2.4.2. Let Γ be an weighted threshold access structure represented by a k-weighted graph G0k of reduced form, k1 = j · 2x and k2 = (j + 1) · 2x− 1, x ≥ 0, j = 2, 3. If k1 ≤ k ≤ k2, then there exists a secret-sharing scheme Σ for the access structure Γ whose average information ratio ARΣ satifies
(32k1+ 2) log2k1+ δ(j)1 k1+ δ0(j)
2 ,l. So em0k can be evaluated recursively as follows.
me0k2 = 2 em02x−1(j+1)−1+ 3 · 2x−1(j + 1) − 1
Hence, the secret-sharing scheme constructed with eΠ0k2 has average informa-tion ratio ARΣ2 = me
0 k2
2k2.
(2) The composition process of G0k1 is shown on the leftmost path of length x from the root. Adding up the orders of all subgraphs involved, we have em0k1 = em0j + em0j−1+Px−1
i=1 me02i·j−1+Px
i=1mM2i−10 j,2ij. Making use of the equation em02x(j+1)−1 = 2x· m0j+ 3x · 2x−1(j + 1) − (2x− 1) from the derivation in (1), we can continue to evaluate em0k1 according to the value of j as follows.
(i) If j = 3, scheme with average information ratio ARΣ1 = me2k0k1
1. The result follows immediately.
Next, we give the expression for emk for a k-weighted graph of general form.
Lemma 2.4.3. Let k = 2x· (j +1)−1, x ≥ 0, j = 2, 3. If emk =Pk occurrences of the vertices in Ai’s and Ci’s in eΠj are exactly the initial values αj,i0 ’s and βj,i0 ’s respectively. For x > 0, recall that Gk is composed of W1 =
than it does in the covering of W1 because it also occurs in M. This is also true for vertices in Ai and Ci, k+12 = 2x−1(j + 1) + 1 ≤ i ≤ k, because all
2 have occurrence one because they only appear in M. Hence, αj,xk+1
2
= βj,xk+1 2
= 1. This proves that the coefficients αxj,i’s and βj,ix ’s satisfy the given recursive relations.
Now, we consider the case when n =Pk
i=1(ai+ ci) is fixed. By evaluating the minimum value of emk, we obtain the lowest possible average information ratio of a secret-sharing scheme constructed with this covering.
Theorem 2.4.4. Let Γ be a weighted threshold access structure represented by a k-weighted graph G = W (a1, . . . , ak, c1, . . . , ck) of order n and k = (j + 1)2x − 1. If ci = 1 for all i 6= k+12 and ai = 1 for all i /∈ T = {1, 2} ∪ {(j + 1)2i|i = 0, 1, . . . , x − 1}. Then
AR(G) ≤ n + 32(k + 1) log2(k + 1) + (δ(j)− 2)k + (δ(j)+ 1) n
where δ(j) is given in Theorem 2.4.2.
Proof. The argument is similar to the proof of Theorem 2.3.5. From the relations given in Lemma 2.4.3, among all the coefficients of ai’s and ci’s, only αxj,i, i ∈ T , and βj,xk+1
2 are equal to one. So emk is minimized if ai = 1 for all i /∈ T and ci = 1 for all i 6= k+12 . We modify the expression for me0k2 in the proof of Theorem 2.4.2 to meet what we need here. In this case, mek= em0k2+P
i∈T ai+ ck+1
2 − (|T | + 1) = em0k+ n − 2k = n +32(k + 1) log2(k + 1)+(δ(j)−2)k +(δ(j)+1). The secret-sharing scheme for this access structure has average information ratio menk.
This result is also very good when k is relatively small compared with n.
The ratio also approaches “1” asymptotically as k fixed. After analyzing the average information ratio produced from each of our constructions separately, we shall give a comparison of them in Section 2.5. For a fair comparison, we consider the same class of k-weighted graphs where k = 3 · 2x − 2. We present the lowest possible average information rate for this class as follows.
Theorem 2.4.5. Let Γ be a weighted threshold access structure represented by a k-weighted graph Gk = W (a1, . . . , ak, c1, . . . , ck) of order n and k = 3 · 2x− 2. If ci = 1 for all i 6= k2 and ai = 1 for all i /∈ T = {1} ∪ {3 · 2i− 1|i = 0, 1, . . . , x − 1}. Then
AR(Gk) ≤ n + (32k + 2) log2(k + 2) − (23 +32log23)k + 23 − 2 log23
n .
Proof. Suppose (Sk
i=1Ai) ∪ (Sk
i=1Ci) is the vertex set of Gk where |Ai| = ai and |Ci| = ci, i = 1, 2, . . . , k. Denote {u} by A0 and {v} by C0. Let (Sk
i=0Ai) ∪ (Sk
i=0Ci) be the vertex set of the (k + 1)-weighted graph Gk+1= W (|A0|, a1, . . . , ak, |C0|, c1, . . . , ck) of order n + 2 where k + 1 = 3 · 2x − 1.
Then Gk+1 satisfies the criteria in Theorem 2.4.4, and the vertex-number sum emk+1 of its covering eΠk+1 is n + 2 +32(k + 2) log2(k + 2) + (δ(2)− 2)(k + 1) + δ(2)+ 1. Now, observe that Gk = Gk+1− (A0 ∪ C0) and the collection of subgraphs obtained from eΠk+1 by deleting u and v from each subgraphs
in eΠk+1 is exactly the complete multipartite covering eΠk of Gk since Gk+1 is
The weighted threshold access structure is a more applicable structure of secret-sharing schemes in reality. In the implementation of such a scheme, the value of k can be thought of as the number of departments or divisions in an organization. In order to have a comparison of the efficiency of our con-structions of secret-sharing scchmes, we let AR1 = 12n+k2+34k−60 log2(k+23 )−32
12n
and AR2 = n+(32k+2) log2(k+2)−(n23+32log23)k+23−2 log23 which are the lowest possi-ble average information ratio derived from our two constructions in Theorem 2.3.5 and Theorem 2.4.5, respectively. Both ratios perform very well when n/k is large. If k is constant, both of them approaches “1” asymptotically.
Let n = µk where µ can be thought of as the average size of departments in the organization. When µ is larger, both AR1 and AR2 become lower for each fixed value of k. Figures 2.3 and 2.4 show the behavior of Morillo’s ratio [28], AR1 and AR2 in the case when µ = 20. As indicated in the figure, AR1 performs better than AR2 when k ≤ 30, whereas AR2 becomes supe-rior to AR1 for all k ≥ 31. Actually, this fact remains true for all values of µ. Therefore, Construction I is more suitable for organizations with fewer departments, whereas Construction II performs especially well for organiza-tions with more departments.
The results in this chpater have been included in the following paper.
”H.-C. Lu and H.-L. Fu, New bounds on the average information rate of secret-sharing schemes for graph-based weighted threshold access structures, Information Sciences, 240 (2013), 83-94.”
(http://dx.doi.org/10.1016/j.ins.2013.03.047)
Figure 2.3: A comparison of the results in the case when µ = 20.
Figure 2.4: A comparison of AR1 and AR2 in the case when µ = 20.
Chapter 3
Optimal Average Information Ratio for Trees
Before taking care of trees, we start this chapter with the introduction of our approach to the determination of the exact values of the optimal average information ratio of graphs of larger girth.
3.1 Our Approach to the Determination of the Exact Values of AR(G)
Let IN(G) = {v ∈ V (G)| degG(v) ≥ 2} and in(G) = |IN(G)|. Given a star covering Π of G with vertex-number sum mΠ, the deduction of Π is defined as dΠ= |V (G)| + in(G) − mΠ. A star covering with the least vertex-number sum gives the largest deduction. We also denote the largest deduction over all star coverings of G as d∗(G), called the deduction of G. A star covering Π with dΠ = d∗(G) is referred to as an optimal star covering of G. The following upper bound on AR(G) is simply a rephrasemant of Theorem 1.3.1 in terms of the deduction of G.
Corollary 3.1.1 ([34]). If Π is a star covering of a graph G with deduction dΠ, then AR(G) ≤ |V (G)|+in(G)−dΠ
|V (G)| .
For the derivation of lower bounds on AR(G), we follow Csirmaz’s
ap-proach stated in Section 1.3.2. Recall that a core of G is a connected subset V0 ⊆ V (G) such that each vertex v ∈ V0 has a designated outside neighbor
¯
v, which refers to a neighbor of v that is outside V0 and is not adjacent to any other vertex in V0, and {¯v|v ∈ V0} is an independent set. In the case of trees, all neighbors of the vertices in a connected set naturally form an independnet set. Therefore a core of a tree can be simplified as a connected subset V0 ⊆ V (G) such that each vertex v ∈ V0 has a designated outside neighbor. In order to cope with the average information ratio, we extend the idea of a core of G. For G 6= K1,1, we define a core cluster of G of size k as a partition C = {V1, V2, . . . , Vk} of IN(G) such that each Vi, i ∈ {1, 2, . . . , k}, is a core of G. The size of a core cluster C is written as cC. We also denote the minimum size of all core clusters of G as c∗(G), called the core number of G. Note that Sk
i=1Vi may not be a core of G, if so, then c∗(G) = 1 for G 6= K1,1. The core number of K1,1 is naturally defined as c∗(K1,1) = 0. A core cluster of size c∗(G) is then called an optimal core cluster of G. The idea of a core cluster helps us establish a lower bound on AR(G).
Theorem 3.1.2. If C is a core cluster of a graph G, then AR(G) ≥ |V (G)| + in(G) − cC
|V (G)| .
Proof. Let C = {V1, V2, . . . , Vk} and Σ be a secret-sharing scheme on G.
Then the function f defined in Section 1.3.2 by the random variables from Σ satisfies inequalities (a) to (e) and Theorem 1.3.9. Since G has no iso-lated vertices, f (v) ≥ 1 for all v ∈ V (G) [13]. We have P
Combining Corollary 3.1.1 and Theorem 3.1.2, we have the following re-sults.
Theorem 3.1.3. The inequality cC ≥ dΠ holds for any star covering Π and core cluster C of a graph G. In particular, c∗(G) ≥ d∗(G).
Corollary 3.1.4. If there exists a star covering Π and a core cluster C of a graph G such that cC = dΠ, then c∗(G) = cC = dΠ = d∗(G) and AR(G) =
|V (G)|+in(G)−c∗(G)
|V (G)| .
As indicated in this result, the equality c∗(G) = d∗(G) makes a criterion for examining whether the lower bound and the upper bound on AR(G) will match. We call G realizable if c∗(G) = d∗(G) holds. In the next section, we shall show that all trees are realizable.
3.2 The Exact Values of the Optimal Infor-mation Ratio of All Trees
Given a tree T , we let IN(T ) and LF (T ) be the sets of all internal vertices and leaves of T respectively. Denote |IN(T )| as in(T ) and |LF (T )| as lf (T ).
Blundo et al.[7] 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 ∈ IN(T ) A(v) ← NT(v) ∩ IN(T )
Π ← φ Π ← Π ∪ {Sv}
Cover(v) E(T ) ← E(T )\E(Sv)
Output the star covering Π V (T ) ← V (T )\((NT(v) ∩ LF (T )) ∪ {v}) for all v′ ∈ A(v) do Cover(v′)
Lemma 3.2.1. Let T be a tree. The star covering Π of T produced by Covering(T ) has deduction dΠ = 1 if T 6= K1,1 and dΠ= 0 if T = K1,1. Proof. For T 6= K1,1, 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 mΠ= lf (T ) + 1 + 2(in(T ) − 1) =
|V (T )| + in(T ) − 1, and we have dΠ= 1.
We shall refine this process and obtain star coverings with higher deduc-tions next.
A vertex v ∈ IN(T ) is called a critical vertex of T if NT(v) ∩ LF (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 . Consider the subgraph HT of T induced byXT and let ΛT (resp.YT) be the set of all nontrivial (resp. trivial) components in HT. Then the set YT is in fact the set of all isolated vertices in HT. So,YT can been seen as a subset of XT. In addition, for any V′ ⊆ V (T ) and E′ ⊆ E(T ), the graph T − V′ is obtained by removing from T all vertices in V′ as well as the 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 3.2.2. Let T 6= K1,1 be a tree. If ΛT = ∅ and |YT| = y ≥ 0, then there exists a star covering Π of T with deduction dΠ = y + 1.
Proof. Let G be an arbitrary component in T −YT. If w1, . . . , wl are 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) ∪ {w1, . . . , wl}. Let H = { ˜G|G is a component in T −YT} and ΠG˜ be the star covering produced by algorithm Covering( ˜G). By the definition ofYT, no ˜G is isomorphic to K1,1, so dΠG˜ = 1 by Lemma 3.2.1. Since S
G∈H˜ E( ˜G) = E(T ), the covering Π =S
G∈H˜ ΠG˜ is a star covering of T with vertex-number sum
mΠ = X
G∈H˜
(|V ( ˜G)| + in( ˜G) − 1)
= V (T ) + X
v∈YT
(degT(v) − 1)
!
+ (in(T ) − y)
− X
v∈YT
degT(v) − (y − 1)
!
= V (T ) + in(T ) − (y + 1).
Next, we consider the core number of T . For a tree T with XT = ∅, {IN(T )} is obviously a core cluster of minimum size. The following lemma is straight forward.
Lemma 3.2.3. Let T 6= K1,1 be 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) ∩ IN(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 form v˜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′, as H+(T, V′, E′).
Observe that, if v ∈ V′ and degT(v) = 2, then v ∈ LF (G+) for exactly two G+’s in the collection H+(T, V′, E′).
Proposition 3.2.4. Let T 6= K1,1 be a tree. If ΛT = ∅ 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 ∈ IN(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 IN(G+i ) 6= ∅. Let Vi = IN(G+i ) ∪ {v|v ∈ V (Gi) ∩YT and degT(v) = 2}. We claim that {V0, V1, . . . , Vy} is a core cluster of T . First, each vertex u ∈ IN(T )\YT belongs to exactly one IN(G+i ) and also exactly one Vi. Each v ∈YT belongs to exactly two G+i ’s. If degT(v) ≥ 3, then v is an internal vertex of one G+i and a leaf of the other. It belongs to exactly one IN(G+i ) and hence exactly one Vi. If degT(v) = 2, then v is a leaf of exactly one component Gi in T − E′ and is a leaf of two subgraphs in H+(T,YT, E′).
Hence it belongs to exactly one Vi and none of the IN(G+j)’s, j = 0, 1, . . . , y.
This shows that {V0, V1, . . . , Vy} is a partition of IN(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 3.2.2 and Corollary 3.1.4.
Before literally proving our main theorem, we examine the relation be-tween the deductions of star coverings of the subtrees in H+(T, V′, E′) and the deduction of a star covering of T more closely.
Lemma 3.2.5. Let V′ be an independent subset of IN(T ) and z = |{v ∈ V′| degT(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 covering ΠT′ of each T′ ∈ H+(T, V′, E′) with deduction dΠT ′, then Π = S
T′∈H+(T,V′,E′)ΠT′ is a star covering of T with deduction dΠ =P
T′∈H+(T,V′,E′)dΠT ′ − z.
Proof. Denote H+(T, V′, E′) as H+ for now. Since S
T′∈H+E(T′) = E(T ), Π is a star covering of T . The vertex-number sum mΠ of Π is
mΠ= X
T′∈H+
(|V (T′)| + in(T′) − dΠT ′)
= |V (T )| + |V′| + in(T ) − (|V′| − z) − X
T′∈H+
dΠT ′
= |V (T )| + in(T ) − X
T′∈H+
dΠT ′ − z
! .
Now, we are in a position to present our main theorem in this chapter.
Theorem 3.2.6. Any tree T is realizable and AR(T ) = n + in(T ) − c∗(T )
n .
Proof. We prove this result by induction on |XT|.
(1) If |XT| = 0 or 1, then ΛT = ∅. The result holds by Proposition 3.2.4.
(2) Suppose that |XT| ≥ 2. By Proposition 3.2.4, we may assume that ΛT 6= ∅. Choose a vertex v ∈ LF (T′) for some T′ ∈ ΛT and let ˜v be the neighbor of v in T′. There are two subtrees G+0 and G+1 in H+(T, {v}, {v˜v}), each of which is not a K1,1. Let G+0 be the one not containing ˜v, then
|XG+
0| < |XT| is obviously true. Since v ∈ LF (G+1), it is no longer a critical vertex of G+1, we also have |XG+1| < |XT|. By induction hypothesis, there exist a star covering Πi of G+i and a core cluster Ci = {Vi1, Vi2, . . . , Viki} with dΠi = cCi = ki > 0, i = 0, 1. Then Π = Π0 ∪ Π1 is a star covering of T . We construct a core cluster of size dΠ next.
(i) If degT(v) ≥ 3, then dΠ = k0 + k1 − 1 by Lemma 3.2.5. Suppose that v ∈ V01. Since V01 is a core of G+0, there is a designated outside neighbor v′ of v in G+0 and outside V01. Now, v′ is an internal vertex of G+0 because v is critical both in T and in G+0. We may assume that v′ ∈ V02. Now, let C = {V01∪ V02, V03, . . . , V0k0, V11, . . . , V1k1}, then
|C| = k0 + k1 − 1. We claim that C is a core cluster of T . First note that IN(G+0) ∪ IN(G+1) = IN(T ) and any two sets in C are disjoint.
Each set in C\{V01∪ V02} is a core of G+0 or G+1, hence a core of T . For V01∪ V02, ˜v is a neighbor of v in T not in V01∪ V02. Since v ∈ LF (T′), v′ is not critical and then has a leaf neighbor v′′ 6= v in G+0 (and in T ) not in V02, so v′′ ∈ V/ 01∪ V02 is the designated outside neighbor of v′ with respect to V01∪ V02, and V01∪ V02 is qualified as a core of T . Therefore, C is a core cluster of T of size dΠ.
(ii) If degT(v) = 2, then dΠ = k0+ k1 by Lemma 3.2.5. Since v is a criti-cal vertex of T , the neighbor v′ 6= ˜v in T is an internal vertex of G+0. We may assume that v′ ∈ V01. Let C = {V01∪{v}, V02, . . . , V0k0, V11, . . . , V1k1}, then |C| = k0+ k1. To show that C is a core cluster of T , it suffices to show that V01∪ {v} is a core of T . Note that v′ is not critical in both G+0 and T . It has a leaf neighbor v′′ 6= v not in V01∪ {v} which serves as a qualified designated outside neighbor of v′ with respect to V01∪ {v}. Besides, ˜v is also a qualified designated outside neighbor of
v with respect to V01∪ {v}. The set V01 ∪ {v} is indeed 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 on AR(T ) coincide. Hence, AR(T ) = n+in(T )−cn ∗(T ).
3.3 The Evaluation of AR(T ) for Some Classes of Trees Using Our Approach
In this section, we evaluate the optimal average information ratio systemat-ically for two infinite classes of trees using our approach.
The only infinite class of trees which has known optimal average informa-tion ratio is the paths. By evaluating the core number, we can easily obtain the known result.
Proposition 3.3.1 ([34]). Let Pn be a path of length n. Then AR(Pn) =
( 3n
2(n+1), if n is even;
3n+1
2(n+1), if n is odd.
Proof. By Proposition 3.2.4, we have c∗(P1) = 0, c∗(P2) = c∗(P3) = 1 and c∗(P4) = 2. Observe that ΛPn = {Pn−4} for all n ≥ 5. Since any leaf of the Pn−4 in ΛPn has degree two in Pn, from the proof of Theorem 3.2.6, we have c∗(Pn) = c∗(Pn−4) + 2. Recursively, we have
c∗(Pn) =
(c∗(Pi) + 2k, if n = 4k + i, i = 1, 2, 3;
c∗(P4) + 2(k − 1), if n = 4k.
= (n
2, if n is even;
n−1
2 , if n is odd.
Hence,
AR(Pn) = (n + 1) + (n − 1) − c∗(Pn)
n + 1 =
( 3n
2(n+1), if n is even;
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.
Theorem 3.3.2. Let Tk be a complete q-ary tree with k levels, q ≥ 2. Then
AR(Tk) =
(qk+2+2qk+1−q2−2q
(q+1)(qk+1−1) , if k is even;
qk+2+2qk+1−q2−q−1
(q+1)(qk+1−1) , if k is odd.
Proof. By Proposition 3.2.4, c∗(T1) = 1 and c∗(T2) = 2. Observe that ΛTk = {Tk−2}, k ≥ 3, 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 Tk
compose a T2, from the proof of Theorem 3.2.6, we get c∗(Tk) = c∗(Tk−2) + qk−2(c∗(T2) − 1) = c∗(Tk−2) + qk−2. Recursively, the core number of Tk can be evaluated as follows.
c∗(Tk) =
(qk−2+ qk−4+ · · · + q2+ c∗(T2), if k is even;
qk−2+ qk−4+ · · · + q + c∗(T1), if k is odd.
=
(qk+q2−2
q2−1 , if k is even;
qk+q2−q−1
q2−1 , if k is odd.
Therefore,
AR(Tk) =
qk+1−1
q−1 +qq−1k−1 − c∗(Tk)
qk+1−1 q−1
=
(qk+2+2qk+1−q2−2q
(q+1)(qk+1−1) , if k is even;
qk+2+2qk+1−q2−q−1
(q+1)(qk+1−1) , if k is odd.
3.4 Concluding Remark
We have proposed the idea of the deduction d∗(G) and the core number c∗(G) of a graph G and showed that these values are the same for any tree
T , thereby proving the upper bound and the lower bound on the optimal average information ratio of a tree coincide. By doing so, we also present a systematic way of evaluating the core number of a tree.
In addition, the condition d∗(G) = c∗(G) makes a criterion for examin-ing whether the upper bound and the lower bound on AR(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. In the next chapter, we study the optimal average information ratio of bipartite graphs of larger 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.
The results in this chapter have been included in the following paper.
”H.-C. Lu and H.-L. Fu, The exact values of the optimal average informa-tion ratio of perfect secret-sharing schemes for tree-based access structures, Designs, Codes and Cryptography (2013), http://dx.doi.org/10.1007/s10623-012-9792-1”
Chapter 4
The Average Information Ratio of Bipartite Graphs
4.1 Some Classes of Realizable Graphs
In this chapter, we need more definitions and notations to facilitate the whole discussion process for bipartite graphs. The girth of G is written as girth(G).
NG(v) denotes the set of all neighbors of v in G and N(S) =S
v∈SNG(v) for any S ⊆ V (G). A vertex v is called a k-vertex of G if degG(v) = k. Let G = (X, Y ) be a bipartite graph with bipartitions X and Y . If H is a subgraph of G, we use XH and YH to denote X ∩ V (H) and Y ∩ V (H) respectively and then H = (XH, YH). In addition, let XH(k) = {x ∈ XH| degH(x) = k}
and XHk+ = {x ∈ XH| degH(x) ≥ k}. The sets YH(k) and YHk+ are defined correspondingly. In the case when H = G, we use X(k) and Xk+ for XG(k) and XGk+ respectively and also use Y(k) and Yk+ for YG(k) and YGk+ respectively for simplicity. In order to have a better description of our approach to the problem regarding bipartite graphs, we give an alternative definition of a core cluster of G. A core cluster g of G is defined as a vertex labeling g : IN(G) → N ∪ {0} such that each g−1(i), i ∈ g(IN(G)), is a core of G.
The size |g(IN(G))| of the clore cluster is denoted as cg in this chapter. The core number of G is still written as c∗(G). As a reminder, for any V′ ⊆ V (G) and any E′ ⊆ E(G), we do not remove resulting isolated vertices from the
subgraphs G − V′ and G − E′. Each isolated vertex is considered as a trivial component in both subgraphs.
As we define an orientation on a specified trail v0 − v1 − · · · − vl (the vi’s may repeat) in the proof of Theorem 4.1.1, “orienting the trail from v0 to vl” means choosing the orientation vi → vi+1 for each edge vivi+1, i = 0, 1, . . . , l − 1, of this trail. For any subgraph H of G, we denote as SvH the star centered at v and having all neighbors of v in H as its leaves. In what follows, we let ΠX(H) = {SxH|x ∈ XH} and ΠY(H) = {SyH|y ∈ YH}.
Both of them are star coverings of H. Unless otherwise specified, a graph G = (X, Y ) always represents a bipartite graph which contains no isolated vertices.
Theorem 4.1.1. Let G = (X, Y ) with |X| ≥ |Y | and girth(G) ≥ 6. If degG(x) ≤ 2 for all x ∈ X, then G is realizable and c∗(G) = |Y2+|.
Proof. Before constructing the desired core cluster, we define an orientation on G first. (i) If G contains a cycle C, then we start with an orientation on C so that C becomes a directed cycle. Next, we repeat the following
Proof. Before constructing the desired core cluster, we define an orientation on G first. (i) If G contains a cycle C, then we start with an orientation on C so that C becomes a directed cycle. Next, we repeat the following