Independent domination in hypercubes

Harary, Frank; Livingston, Marilynn

doi:10.1016/0893-9659(93)90027-k

Cited by 27 publications

(17 citation statements)

References 4 publications

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“…Theorem 4: An n-variable Boolean function R(n) is a root function if and only if R(n) corresponds to an independent dominating set in a hypercube of dimension n. Thus, Q ind (G) represents the number of true minterms in a min-root function. It has been reported [17] that Q ind (G) is 1, 2, 2 4, 8, 12, for n = 1, 2, 3, 4, 5, 6 respectively, which match with our findings. However, till date not much is known about Q ind (G) for higher values of n. Also, enumeration of total number of root functions (or the number of independent dominating sets in a hypercube) is not yet studied.…”

Section: Independent Domination In Hypercubessupporting

confidence: 93%

“…In this paper, we introduce a new class of Boolean functions, called as root functions, and show that no such function can appear as a faulty response in any two-level irredundant AND-OR circuits under stuck-at faults. We prove that set of all root functions and the set of their possible faulty response functions are not only mutually exclusive but their union also collectively exhausts all 2 Boolean functions of n variables for any n. More interestingly, we show that the number of such root functions is exactly equal to the number of independent domination in a hypercube of dimension n [17].…”

Section: Introductionmentioning

confidence: 86%

“…In this section, we show that the set of n-variable root functions has a one-to-one correspondence to the independent dominating sets in a hypercube of dimension n [17,18].…”

Section: Relation To Independent Dominating Setmentioning

confidence: 97%

See 2 more Smart Citations

Inadmissible Class of Boolean Functions under Stuck-at Faults

Das

Chowdhury

Bhattacharya

et al. 2014

2014 IEEE 44th International Symposium on Multiple-Valued Logic

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Many underlying structural and functional factors that determine the fault behavior of a combinational network, are not yet fully understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in irredundant two-level circuits even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response from an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. We report some bounds and enumerate the total number of root functions up to 6 variables. Finally, we point out several open problems and possible applications of root functions in logic design and testing.

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Section: Independent Domination In Hypercubessupporting

confidence: 93%

Section: Introductionmentioning

confidence: 86%

See 1 more Smart Citation

Inadmissible Class of Boolean Functions under Stuck-at Faults

Das

Chowdhury

Bhattacharya

et al. 2014

2014 IEEE 44th International Symposium on Multiple-Valued Logic

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“…The first assertion of Theorem 2.1 is based on the fact that hypercubes Q 2 k −1 contain perfect codes, cf. [7]. Since the domination number of a graph with a perfect code is equal to the size of such a code, the assertion follows.…”

Section: Preliminariesmentioning

confidence: 84%

(Total) Domination in Prisms

Azarija¹,

Henning²,

Klavžar³

2017

Electron. J. Combin.

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With the aid of hypergraph transversals it is proved that γ t (Q n+1 ) = 2γ(Q n ), where γ t (G) and γ(G) denote the total domination number and the domination number of G, respectively, and Q n is the n-dimensional hypercube. More generally, it is shown that if G is a bipartite graph, then γ t (G K 2 ) = 2γ(G). Further, we show that the bipartite condition is essential by constructing, for any k ≥ 1, a (non-bipartite) graph G such that γ t (G K 2 ) = 2γ(G) − k. Along the way several domination-type identities for hypercubes are also obtained.

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“…Denote by K q (n, 1, 2) the minimal cardinality of a code C ⊂ Q n with covering radius 1 and minimum distance 2. Well-known results are K q (2, 1, 2) = q and K q (3, 1, 2) = q 2 /2 as well as K 2 (4, 1, 2) = 4, K 2 (5, 1, 2) = 8, K 2 (6, 1, 2) = 12, K 3 (4, 1, 2) = 9, K 4 (4, 1, 2) = 28 and K q (n + 1, 1, 2) ≤ q • K q (n, 1, 2), see [4,3,8,6].…”

Section: Introductionmentioning

confidence: 98%

On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

Haas¹,

Quistorff²

2007

Electron. J. Combin.

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Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

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Independent domination in hypercubes

Cited by 27 publications

References 4 publications

Inadmissible Class of Boolean Functions under Stuck-at Faults

Inadmissible Class of Boolean Functions under Stuck-at Faults

(Total) Domination in Prisms

On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

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