Repository Universitas Sanata Dharma

Santoso, Eko Budi

 
On Independent [1, 2]-sets in Hypercubes 
Eko Budi Santoso1*, Reginaldo M. Marcelo2, and Mari-Jo P. Ruiz2 
1Department of Mathematics Education, Sanata Dharma University, Indonesia 
2Department of Mathematics, Ateneo de Manila University, Philippines 
Abstract. Given a simple graph G, a subset 𝑆 ⊆ 𝑉(𝐺) is an independent [1, 
2]-set if no two vertices in S are adjacent and for every vertex 𝑣 ∈  𝑉(𝐺)\𝑆, 
1 ≤  |𝑁(𝑣)  ∩  𝑆 |  ≤  2, that is, every vertex 𝑣 ∈  𝑉(𝐺)\𝑆 is adjacent to at 
least one but not more than two vertices in S. This paper investigates the 
existence of independent [1, 2]-sets of hypercubes. We show that for some 
positive integer k, if 𝑛 = 2𝑘 − 1, then hypercubes 𝑄𝑛 and 𝑄𝑛+1 have an 
independent [1, 2]-set. Furthermore, for 1 ≤  𝑛 ≤  4, we find bounds for 
the minimum and maximum cardinality of an independent [1, 2]-set of 
hypercube 𝑄𝑛, while for 𝑛 =  5, 6, we get the maximum of cardinality of an 
independent [1, 2]-set of hypercube 𝑄𝑛.  
1 Introduction 
Let 𝐺 be a simple graph, that is, it is an undirected graph, has no loop, and has no multiple 
edges. The open neighborhood of a vertex 𝑣 ∈  𝑉(𝐺) is the set 𝑁(𝑣)  =  {𝑢|𝑢𝑣 ∈  𝐸(𝐺)} of 
vertices adjacent to 𝑣. Each vertex in 𝑢 ∈  𝑁(𝑣) is called a neighbor of 𝑣 and the degree of 
𝑣 is 𝑑(𝑣)  =  |𝑁(𝑣)|. For a set 𝑆 and a vertex 𝑣, we denote the number of neighbors of 𝑣 in 
𝑆 as 𝑑𝑆(𝑣), that is, 𝑑_𝑆 (𝑣)  =  |𝑁(𝑣)  ∩  𝑆 |. A set 𝑆 is independent if no two vertices in 𝑆 
are adjacent and dominating if every vertex not in 𝑆 is adjacent to some vertices in 𝑆. 
Chellali et al., in [1], define a subset 𝑆 ⊆  𝑉(𝐺) to be a [𝑗, 𝑘]-set if for every vertex 𝑣 ∈
 𝑉(𝐺)\𝑆, 𝑗 ≤  𝑑𝑆 (𝑣)  ≤  𝑘, that is, every vertex in 𝑉(𝐺)\𝑆 is adjacent to at least 𝑗 vertices, 
but not more than 𝑘 vertices in 𝑆. For 𝑗 =  1, a [1, 𝑘]-set 𝑆 is a dominating set, since every 
vertex in 𝑉(𝐺)\𝑆 has at least one neighbor in 𝑆 (is dominated by 𝑆). The major focus in this 
study is finding bounds on the minimum cardinality of a [1, 2]-set [1]–[4]. 
In [5], Chellali et al. continue the study of [𝑗, 𝑘]-sets and add the requirement that the sets 
be independent. A dominating set 𝑆 is an independent [1, 𝑘]-set of 𝐺 if 𝑆 is independent and 
1 ≤ 𝑑𝑆(𝑣) ≤ 𝑘 for all 𝑣 ∈ 𝑉(𝐺)\𝑆. In this paper, we will exclusively focus on independent 
[1, 2]-set. Given a graph 𝐺, we denote by 𝑖[1,2](𝐺) the minimum cardinality of an independent 
[1, 2]-set of 𝐺 and by 𝛼[1,2](𝐺) the maximum cardinality of an independent [1, 2]-set of 𝐺. 
Unfortunately, not every graph has an independent [1, 2]-set. Thus, beside finding the lower 
and upper bounds cardinality of an independent [1, 2]-set of a graph, investigating the 
existence of an independent [1, 2]-set for some graphs is another focus in this study [6], [7]. 
In this work, we investigate the existence of independent [1, 2]-sets of hypercube 𝑄𝑛. 
Moreover, we find bounds for the minimum and maximum cardinality of an independent 
 
* Corresponding author: ekobudisantoso@usd.ac.id 
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons 
Attribution License 4.0 (https://creativecommons.org/licenses/by/4.0/).
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[1, 2]-set of hypercube 𝑄𝑛, for 𝑛 =  1, 2, 3, and 4. For 𝑛 =  5, 6, we get bounds for the 
maximum cardinality of an independent [1, 2]-set of hypercube 𝑄𝑛. 
The 𝑛-cube or 𝑛-dimensional hypercube 𝑄𝑛 is defined recursively in terms of the 
Cartesian product of two graphs as follows [8]: 
𝑄1 = 𝐾2 (a complete graph of order 2) 
𝑄𝑛 = 𝐾2□𝑄𝑛−1  
 
 
Fig. 1. Hypercubes (a) 𝑄1, (b) 𝑄2, and (c) 𝑄3 with an independent [1, 2]-set. 
The hypercube of dimension 𝑛 may also be defined as a graph with vertex set 𝑉(𝑄𝑛) the 
set of all binary 𝑛-tuples of zeros and ones and set 𝐸(𝑄𝑛) the set of pairs of vertices 𝑢 =
𝑢1𝑢2. . . 𝑢𝑛 and 𝑣 = 𝑣1𝑣2. . . 𝑣𝑛, where ∑ |𝑢𝑖 − 𝑣𝑖|
𝑛
𝑖=1  =  1. In this representation, two vertices 
of 𝑄𝑛 are adjacent if and only if their binary 𝑛-tuples differ in exactly one place. Fig. 1 
illustrates hypercubes 𝑄1, 𝑄2, and 𝑄3. 
 
Observation 1 The sets 𝑆1 = {0} and 𝑆2 = {00,11} are independent [1, 2]-sets of 𝑄1 and 
𝑄2, respectively. Furthermore, 𝑖[1,2](𝑄1) = 𝛼[1,2](𝑄1) = 1, 𝑖[1,2](𝑄2) = 𝛼[1,2](𝑄2) = 2, and 
𝑆1 is an efficient dominating set of 𝑄1. 
 
Observation 2 The set 𝑆3 = {000,111} is an independent [1, 2]-set of 𝑄3. No singleton 
subset of 𝑉(𝑄3) can be a dominating set and any independent set of cardinality three is not 
a [1, 2]-set; hence 𝑖[1,2](𝑄3) = 𝛼[1,2](𝑄3) = 2. Furthermore, 𝑆3 is an efficient dominating set 
of 𝑄3. The independent [1, 2]-set 𝑆3 is not unique. 
 
In Observations 1 and 2 we noted that hypercubes 𝑄1 and 𝑄3 have efficient dominating 
sets. The study of the existence of efficient dominating sets in 𝑄𝑛 has been done in the context 
of single error-correcting codes [9]. 
 
Theorem 1 (Livingston [10]). An 𝑛-dimensional hypercube 𝑄𝑛 has an efficient dominating 
set if and only if 𝑛 = 2𝑘 − 1, for some positive integer 𝑘. 
2 Main Result  
In the following discussion, we show that hypercubes 𝑄𝑛, for 𝑛 =  4, 5, and 6, have an 
independent [1, 2]-set. 
 
Proposition 1. 𝑄4 has an independent [1, 2]-set. Furthermore, 𝑖[1,2](𝑄4) = 𝛼[1,2](𝑄4) = 4. 
 
Proof. We prove the proposition by construction. We will construct an independent [1, 2]-
set 𝑆4 using 𝑆3 in Observation 2. 
Since by definition, 𝑄4 = 𝐾2□𝑄3, as illustrated in Fig. 2, we may consider 𝑄4 as formed 
from two copies of 𝑄3, say 𝐴 and 𝐵, respectively. We label the vertices of 𝑄4 using vertex 
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labeling of 𝑄3 with prefix 0 added in the vertices in 𝐴, and 1 for the vertices in 𝐵. Thus if 
𝑣1𝑣2𝑣3 ∈ 𝑉(𝑄3), then its corresponding vertices in 𝑄4 will be 0𝑣1𝑣2𝑣3 and 1𝑣1𝑣2𝑣3. 
 
 
Fig. 2. The hypercubes 𝑄4 with an independent [1, 2]-set. 
We form 𝑆4 by taking a copy of 𝑆3 in 𝐴, and another copy in 𝐵. For example, 𝑆4 =
{0000,0111,1001,1110}. It follows that 𝑖[1,2](𝑄4) = 4 since a vertex subset with at most 3 
elements can only dominate 12 vertices at most, while |𝑉(𝑄4)| = 16. Finally, 𝛼[1,2](𝑄4) =
4 since an independent [1, 2]-set with 5 elements will have at least 3 elements in 𝐴 or 𝐵, 
contradicting 𝛼[1,2](𝑄3) = 2. □ 
 
We note that the independent [1, 2]-set of 𝑄4 is not unique. Fig. 3 shows another 
independent [1, 2]-set of 𝑄4. The advantage of the construction in the proof of Proposition 1 
is that we use the independent [1, 2]-set of 𝑄3 to construct an independent [1, 2]-set of 𝑄4. 
Using a similar technique, we construct an independent [1, 2]-set of 𝑄5 and 𝑄6. Also, we 
observe in Fig. 2, that 𝑆4 is not an independent [1, 1]-set. Furthermore, some vertices in 𝑄4, 
namely vertices 0001, 0110, 1000, 1111, are adjacent to two elements of 𝑆4. We need to 
consider such vertices in the construction of 𝑆5. 
 
 
Fig. 3. Another independent [1, 2]-set for 𝑄4. 
 
Proposition 2. 𝑄5 has an independent [1, 2]-set and 𝛼[1,2](𝑄5) = 8. 
 
Proof. We prove the proposition by construction. We will construct an independent [1, 2]-
set 𝑆5 using 𝑆4 in Proposition 1. We consider 𝑄5 as formed from two copies of 𝑄4, say 𝐴 and 
𝐵, respectively. We label the vertices of 𝑄5 using vertex labeling of 𝑄4 with prefix 0 added 
in the vertices in 𝐴, and 1 for the vertices in 𝐵. 
As illustrated in Fig. 4, we form 𝑆5 by taking a copy of 𝑆4 in Proposition 1 for 𝐴. We 
consider another independent [1,2]-set in 𝐵 such that the union with 𝑆4 is an independent of 
𝑄5. For example, 𝑆5 = {00000,00111,01001,01110,10101,10010,11100,11011}. It 
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follows that 𝛼[1,2](𝑄5) = 8 since an independent [1, 2]-set with 9 elements will have at least 
5 elements in 𝐴 or 𝐵, contradicting 𝛼[1,2](𝑄4) = 4.  □ 
 
 
Fig. 4. The hypercube 𝑄5 with an independent [1, 2]-set. 
 
Proposition 3. 𝑄6 has an independent [1, 2]-set and 𝛼[1,2](𝑄6) = 16. 
 
Proof. Using similar technique as in Propositions 1 and 2, we construct an independent [1, 2]-
set 𝑆6 using 𝑆5. We consider 𝑄6 as formed from two copies of 𝑄5, say 𝐴 and 𝐵, respectively. 
We use 𝑆5 in Proposition 2 as an independent [1,2]-set for A. As illustrated in Fig. 5, we 
construct an independent [1,2]-set for B such that the union with 𝑆5 is an independent [1,2]-
set of 𝑄6. 
𝑆6 = {000000,000111,001001,001110,010101,010010,011100,011011, 
100100, 100011, 101101, 101010, 110001, 110110, 111000, 111111} 
is an independent [1, 2]-set of 𝑄6. It follows that 𝛼[1,2](𝑄6) = 16 since an independent [1, 2]-
set with 17 elements will have at least 9 elements in 𝐴 or 𝐵, contradicting 𝛼[1,2](𝑄5) = 8. □ 
 
Theorem 2 If 𝑛 = 2𝑘 − 1, for some positive integer 𝑘, then 𝑄𝑛 and 𝑄𝑛+1 have an 
independent [1, 2]-set. 
 
Proof. By proposition 1, 𝑄𝑛 with 𝑛 = 2
𝑘 − 1 has an efficient dominating set, that is, an 
independent [1, 1]-set, say 𝑆𝑛. Then 𝑆𝑛 is also an independent [1, 2]-set. If 𝐴 =
{𝑎1𝑎2𝑎3 . . . 𝑎𝑛𝑎𝑛+1 ∈ 𝑉(𝑄𝑛+1)|𝑎1 = 0}, then T = {0s1s2s3. . . sn |s1s2s3 . . . sn ∈ Sn} is an 
independent [1, 1]-set of Q𝑛+1[A]. We observe that 
𝑊 =  {𝑢1𝑢2𝑢3 . . . 𝑢𝑛 ∈ 𝑉(𝑄𝑛)|𝑢𝑘 = 𝑠𝑘 , 1 ≤  𝑘 ≤  𝑛 −  1, and |𝑢𝑛  −  𝑠𝑛|  =  1, 
for all 𝑠1𝑠2𝑠3 . . . 𝑠𝑛  ∈  𝑆𝑛}  
is also an independent [1, 1]-set of 𝑄𝑛. Let 𝐵 = {𝑏1𝑏2𝑏3 . . . 𝑏𝑛𝑏𝑛+1  ∈  𝑉(𝑄𝑛 + 1) | 𝑏1 =
 1}. Then a set 𝑈 = {1𝑢1𝑢2𝑢3 . . . 𝑢𝑛 |𝑢1𝑢2𝑢3 . . . 𝑢𝑛 ∈ 𝑊} is an independent [1, 1]-set of 
𝑄𝑛+1[B]. 
 
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Fig. 5. The hypercube 𝑄6 with an independent [1, 2]-set. To avoid a confusion, we only represent some 
edges from vertices 0𝑣2𝑣3𝑣4𝑣5𝑣6 to 1𝑣2𝑣3𝑣4𝑣5𝑣6. 
 
Now we consider Sn+1  =  T  ∪  U. First we note that 𝑆𝑛+1 is a dominating set of 𝑄𝑛+1 
since sets 𝑇 and 𝑈 are independent [1, 1]-sets of 𝑄𝑛+1[𝐴] and 𝑄𝑛+1[𝐵], respectively, where 
sets 𝐴 and 𝐵 are disjoint sets with union 𝑉(𝑄𝑛+1). Let 𝑣 = 𝑣1𝑣2𝑣3. . . 𝑣𝑛+1 and 𝑤 =
𝑤1𝑤2𝑤3. . . 𝑤𝑛+1 be elements of 𝑆𝑛+1. If 𝑣1 = 𝑤1 then they are not adjacent since either both 
of them are elements of 𝑇 or 𝑈. 
Suppose 𝑣1 ≠ 𝑤1. Then one of them is an element of 𝑇 and the other one is an element 
of 𝑈. We recall the construction of 𝑇 and 𝑈. If 0𝑣2𝑣3. . . 𝑣𝑛𝑣𝑛+1 is an element of 𝑇 then vertex 
1𝑣2𝑣3. . . 𝑣𝑛𝑘 with 𝑘 = |𝑣𝑛+1 − 1| is an element of 𝑈. Conversely if 1𝑤2𝑤3. . . 𝑤𝑛𝑤𝑛+1 is an 
element of 𝑈, then vertex 0𝑤2𝑤3. . . 𝑤𝑛𝑘 with 𝑘 = |𝑤𝑛+1 − 1| is an element of 𝑇. Thus, if 
𝑣1𝑣2𝑣3. . . 𝑣𝑛+1 and 𝑤1𝑤2𝑤3. . . 𝑤𝑛+1 are elements of 𝑆𝑛+1 with 𝑣1 ≠ 𝑤1, then ∑ |𝑣𝑖 −
𝑛+1
𝑖=1
𝑤𝑖| ≥ 2. Hence they are not adjacent. 
We have shown that 𝑆𝑛+1 is an independent dominating set. We need to show that it is a 
[1, 2]-set. Let 𝑡 ∈  𝐴 or 𝑡 ∈  𝐵. In any case, 𝑡 is adjacent to at most one element in 𝑇 and at 
most one element in 𝑈. So, 𝑡 is adjacent to at most two elements of 𝑆_{𝑛 + 1}. By similar 
argument, if 𝑡_1 =  1, then 𝑡 is adjacent to at most two elements of 𝑆𝑛+1, and it follows that 
𝑆𝑛+1 is an independent [1, 2]-set of 𝑄𝑛+1.  □ 
3 Conclusion and Remarks 
This paper has shown the existence of an independent [1, 2]-set of hypercube 𝑄𝑛 for 1 ≤
 𝑛 ≤  6 and 𝑛 =  2𝑘, for some positive integer 𝑘. For further study, one may investigate the 
existence of an independent [1, 2]-set of hypercube 𝑄𝑛 for all positive integer 𝑛 together with 
the bounds of minimum and maximum cardinality of an independent [1, 2]-set.  
 
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On Independent [1, 2]-sets in Hypercubes

Sarwono, Robertus Budi

a.n. MENTERI HUKUM
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Direktur Hak Cipta dan Desain Industri
Agung Damarsasongko,SH.,MH.
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Digital Financial Literacy dan Financial Inclusion dengan
Moderasi Non-Consumptive Behavior
Tanggal dan tempat diumumkan untuk pertama
kali di wilayah Indonesia atau di luar wilayah
Indonesia
:
21 Agustus 2025, di Kota Yogyakarta
Jangka waktu pelindungan : Berlaku selama hidup Pencipta dan terus berlangsung selama 70 (tujuh
puluh) tahun setelah Pencipta meninggal dunia, terhitung mulai tanggal
1 Januari tahun berikutnya.
Nomor Pencatatan : 000964203
adalah benar berdasarkan keterangan yang diberikan oleh Pemohon.
Surat Pencatatan Hak Cipta atau produk Hak terkait ini sesuai dengan Pasal 72 Undang-Undang Nomor 28 Tahun 2014 tentang
Hak Cipta.
LAMPIRAN PENCIPTA
No Nama Alamat
1 Christina Heti Tri Rahmawati Purworejo RT 008/ RW 002, Dlingo, Mojosongo, BoyolaliMojosongo, Kab. Boyolali
2 Trisnawati Rahayu Jl. Cik Di Tiro No. 2 RT 059 / RW 012 Terban, Gondokusuman, YogyakartaGondokusuman, Kota Yogyakarta
3 Ignatius Soni Kurniawan Dusun Pepe RT 004 / RW 002 Muntilan, Muntilan, MagelangMuntilan, Kab. Magelang
4 Syamsul Hadi, S.E., M.M. Nglarang, Malangrejo RT 005 / RW 035, Wedomartani, Ngemplak, SlemanNgemplak, Kab. Sleman
5 Jesica Pradana Koesnadi Serangan NG II/73.A RT 008 / RW 002 Notoprajan, Ngampilan, YogyakartaNgampilan, Kota Yogyakarta
Nomor Pencatatan
000964203
Tgl. Pencatatan
2025-08-21
Status
Diterima
Entrepreneurial Resilience Pelaku UMKM Perempuan: Peran Digital Financial
Literacy dan Financial Inclusion dengan Moderasi Non-Consumptive Behavior
Nomor Permohonan
EC002025123942
Tanggal Permohonan
2025-09-03
Tanggal Pertama Kali Diumumkan
2025-08-21
Tanggal Berakhir Masa Pelindungan
-
Uraian Ciptaan Kesenjangan gender menjadi hambatan memulai usaha, walaupun UMK
M di Indonesia sebagian besar dikelola perempuan. Selain itu, belum se
mua pelaku UMKM perempuan mempunyai pemahaman baik terkait digi
tal financial literacy dan dapat mengakses layanan keuangan formal ser
ta masih rendahnya kesadaran untuk tidak berperilaku konsumtif agar t
erhindar investasi ilegal. Oleh karena itu, dibutuhkan atribut non-consum
ptive behavior sehingga dapat mengendalikan diri membeli barang sesu
ai kebutuhan untuk meningkatkan entrepreneurial resilience. Hasil penel
itian digital financial literacy berpengaruh terhadap entrepreneurial resili
ence, digital financial literacy berpengaruh terhadap entrepreneurial resi
Kembali ke pencarian
Permohonan dengan nama
pemilik yang sama dengan
Sentra HKI Universitas Sanata
Dharma
Perilaku Kesehatan (Health Behavior):
Aplikasi Teori Perilaku untuk Promosi
Kesehatan
Diterima EC00202114094
Seri Alamku, Sayang: Kering Kemarau
Diterima EC00202115283
Seri Alamku, Sayang: Menyambut
Mentari
Diterima EC00202115285
Seri Alamku, Sayang: Pesta
Menyambut Hujan
Diterima EC00202115310
Pendidikan ala Warung Pojok.
Catatan-catatan Prof. Dr. N. Driyarkara,
Privacy  - Terms
Pangkalan Data Kekayaan Intelektual
Pencarian Data EC002025123942Hak Cipta Normal
Penelusuran Teks Penelusuran Gambar Total Rate Us Panduan
11/15/25, 11:00 PM Pangkalan Data Kekayaan Intelektual
https://pdki-indonesia.dgip.go.id/detail/e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 1/3
lience dimediasi financial inclusion dan dimoderasi non-consumptive be
havior pelaku UMKM perempuan di DIY.
Jenis Ciptaan Poster
Pemegang
Pencipta
Konsultan / Kuasa
Data per November 15, 2025
S.J. tentang masalah Sosial, Politik
dan Budaya
Diterima EC00202116577
Mata Air, Air Mata Kota
Diterima EC00202116579
Hilangnya Halaman Rumahku
Diterima EC00202117375
Nietzsche
Diterima EC00202121741
Opera Tanpa Kata
Diterima EC00202121743
Asyiknya Permainan Tradisional untuk
Mempertajam Hati Nurani Anak (Seri
2)
Diterima EC00202122086
Disclaimer: Direktorat Jenderal Kekayaan Intelektual tidak menjamin keakuratan informasi yang terdapat dalam Pangkalan Data Kekayaan Intelektual ini. Pembaharuan, Koreksi, atau
perubahan terkini mungkin tidak disertakan. Disarankan untuk berkonsultasi dengan Konsultan kekayaan Intelektual terdaftar jika diperlukan pencarian menyeluruh terhadap merek
Nama Kewarganegaraan
Sentra HKI Universitas Sanata Dharma ID
Nama Kewarganegaraan
Christina Heti Tri Rahmawati ID
Trisnawati Rahayu ID
Ignatius Soni Kurniawan ID
Syamsul Hadi, S.E., M.M. ID
Jesica Pradana Koesnadi ID
Nama
Sentra HKI Universitas Sanata Dharma
11/15/25, 11:00 PM Pangkalan Data Kekayaan Intelektual
https://pdki-indonesia.dgip.go.id/detail/e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 2/3
HASIL PENELITIAN
DAFTAR PUSTAKA


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