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ABSTRACT
An outer-independent k-rainbow dominating function of a graph G is a function f from to the set of all subsets of
such that both the following hold: (i)
whenever v is a vertex with
, and (ii) the set of all
with
is independent. The outer-independent k-rainbow domination number of G is the invariant
, which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it.
1. Introduction
In general, we follow the notation and graph theory terminology in [Citation1]. Specifically, let be a finite simple graph. For any vertex u in G, the open neighbourhood of u, written
, is the set of vertices adjacent to u and the closed neighbourhood of u is the set
. The degree of a vertex
is
. The minimum and maximum degrees of G are denoted by
and
, respectively. A leaf is a vertex of degree one, and a support vertex is a vertex adjacent to a leaf. We denote the sets of all leaves and all support vertices of G by
and
, respectively. For a vertex
, the set of leaf neighbours of v is denoted by
. If
, then
(respectively,
) denotes the union of open (closed) neighbourhoods of all vertices of A. (If the graph G under consideration is not clear we write
, and so on.) We denote by
and
the path and cycle on n vertices, respectively. The distance between two vertices u and v in a connected graph G is the length of a shortest uv-path in G. The diameter of a graph G, denoted by
, is the greatest distance between two vertices of G. For a vertex v in a rooted tree T, let
and
denote the set of children and descendants of v, respectively and let
. Also, the depth of v,
, is the largest distance from v to a vertex in
. The maximal subtree at v is the subtree of T induced by
, and is denoted by
. By
we denote a complete bipartite graph with partite sets of cardinalities p and q. A star is a
and a double star
, where
, is a tree containing exactly two non-leaf vertices which one is adjacent to p leaves and the other is adjacent to q leaves. By
we denote the induced subgraph of a graph G with vertex set
.
A set is independent if no two vertices in I are adjacent. The maximum cardinality of an independent set in G equals the independence number
. A vertex cover of a graph G is a set of vertices that covers all the edges. The minimum cardinality of a vertex cover is denoted by
. The following theorem due to Gallai.
Theorem 1.1
[Citation2]
Let G be a graph. A subset I of is independent if and only if
is a vertex cover of G. In particular,
.
A set in G is called a dominating set if
. The domination number
equals the minimum cardinality of a dominating set in G. For many applications, it is not possible to use an arbitrary dominating set D of G. One possible form of restriction is based on imposing some conditions on the set
. Here we concentrate on the property of being outer-independent, i.e.
is independent. Results on outer-independent domination parameters can be found e.g. in [Citation3–7].
For a positive integer k we denote the set by
. The power set (that is, the set of all subsets) of
is denoted by
. Let G be a graph and let f be a function that assigns to each vertex a subset of
; that is,
. The weight,
, of f is defined as
. The function f is called a k-rainbow dominating function (a kR-function) on G if for each vertex
with
the condition
is fulfilled. Given a graph G, the minimum weight of a k-rainbow dominating function is called the k-rainbow domination number of G, which we denote by
. The concept of rainbow domination was introduced in [Citation8] and has been studied extensively [Citation9–15].
Here we introduce and study a new variant of a k-rainbow dominating function. A k-rainbow dominating function is an outer-independent k-rainbow dominating function (an OIkRD-function) on G if the set
is independent. The outer-independent k-rainbow domination number
is the minimum weight of an OIkRD-function on G. An OIkRD-function of weight
is called a
-function. Since any OIkRD-function is a kRD-function, we have
(1)
(1) In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it.
2. Preliminary results
In this section, we present basic properties of outer-independent k-rainbow domination number . We begin with three simple observations.
Observation 2.1
If are all components of a graph G, then
.
Observation 2.2
For any graph G of order n, . In particular,
whenever
.
Observation 2.3
For any OIkRD-function f on a graph G, the set is a vertex cover of G. Hence
, where
is the set of all isolated vertices of G. In particular,
.
Notice also that the outer-independent domination is the same as the outer-independent 1-rainbow domination if we view an outer-independent dominating set D as an outer-independent 1-rainbow dominating function f defined by when
and
otherwise. Therefore here we concentrate on the case when a graph G is connected and
.
Now we characterize all connected graphs of order attaining the lower bound in Observation 2.2.
Theorem 2.1
Let be a positive integer and let G be a connected graph of order
. Then
if and only if
, where
is a graph of order
.
Proof.
First assume that , where
is a graph of order
. Define a function
by
for
and
for
in such way that
and
. Obviously, f is an OIkRD-function on G with
. Therefore
.
Conversely, assume that . Let f be a
-function on G. Since
, there exists a vertex v with
. This implies that
and
for all
. Moreover,
. Hence
for all
. Since v was chosen arbitrarily,
, where a graph
has no edges.
Theorem 2.2
Let be a positive integer and let G be a connected graph of order
. Then
if and only if the following holds:
there is no h-order graph
,
, such that
.
there exist two nonempty disjoint vertex sets A, B such that: (i)
and
, (ii) every vertex of
is adjacent to every vertex of
except at most one vertex in B, (iii)
is independent, and (iv) for each
,
or
.
Proof.
First assume that G satisfies (I) and (II). It follows from Theorem 2.1 and (I) that . Now define the function
by
for
,
for
, and
for
such that
and
when
and by
for
,
for
, and
for
such that
and
when
. Obviously, f is an OIkRD- function of G with
and thus
.
Conversely, assume that . It follows from Theorem 2.1 that G satisfies (I). Now we show that G satisfies (II). Let f be a
-function on G. Choose f so that
is as small as possible. Since
, there exists a colour, say k, which appears exactly twice and each other colour appears exactly once. Hence there are two vertices, say z and w, such that
. Since
, there exists a vertex v with
which implies that
and that
for each
.
Now let and
. Since
, we have
. On the other hand, if
, then the function g defined by
and
otherwise, is an OIkRD-function of G with weight less that
which is a contradiction. Thus A and B are non-empty sets. Since f is an OIkRD-function, the set
is independent, that is G satisfies (iii).
It follows from that
and so (i) holds. Since for each vertex
,
has a colour which is not appeared in other vertices, every vertex in
is adjacent to every vertex of A. Also since the colour k appears exactly in
and
, each vertex of
must be adjacent to one of the vertices z and w. Hence (ii) holds.
Finally, if for some ,
and
, then the function g defined on G by
,
for some
and
otherwise, is a
-function which contradicts the choice of f. Thus (iv) holds and the proof is complete.
Proposition 2.1
For any graph G of order n, . If G has no isolated vertices, then
.
Proof.
Let be a
-function on G,
. Define the function
as follows:
when
,
when
, and
otherwise. Clearly
is an OIsRD-function and so
.
If C is a minimum vertex cover of G, then the function defined by
when
and
otherwise, is an OI1RD-function on G with weight
. The equality
now follows by Observation 2.3.
Finally, the right side inequality follows by Observation 2.2.
Proposition 2.2
For any graph G, . If
then
.
Proof.
Let C be any minimum vertex cover set of G and define the function by
for
,
when
, and
otherwise. Clearly f is an OIkRD-function on G which immediately implies the required.
The bounds in Proposition 2.2 are attainable. Let G be a graph such that each vertex is either a leaf or a support vertex and let each support vertex of G is adjacent to at least k + 1 leaves. Then clearly is a minimum vertex cover set and the function
defined as
when u is a support vertex and
when u is a leaf, is an OIkRD-function on G of minimum weight. Thus
.
We will say that a graph G is a vertex cover outer independent k-rainbow graph, a VCOI-k-rainbow graph for short, if .
Proposition 2.3
A graph G with no isolated vertex, is VCOI-k-rainbow if and only if it has a -function f such that for each vertex x, either
or
.
Proof.
Assume that G is a VCOI-k-rainbow graph and let D be a minimum vertex cover set of G. Then the function defined by
for
and
otherwise, is an OIkRD-function on G which implies that
. Thus all inequalities in this chain must be equality and so
, yielding f is a
-function satisfying that for each vertex x either
or
.
Conversely, assume that there exists a -function h such that for each vertex x, either
or
. Since the set
is a vertex cover set of G, we have
. By Proposition 2.2 we deduce that
and this implies that G is a VCOI-k-rainbow graph.
Proposition 2.4
Let H be an induced subgraph of a graph G. Then .
Proof.
Let f be a -function on H. Define an OIkRD-function h on G as follows:
when
and
otherwise. Since
, we have the desired inequality.
Observation 2.4
Let f be an OIkRD-function on a graph G and for each
. Then
.
Theorem 2.3
Let G be a graph of order at least two and . Then
.
Proof.
Let f be a -function on G, and
,
. Assume without loss of generality that
.
Define by
when
and
otherwise. Clearly g is an OIkRD-function on G. This fact and Observation 2.4 lead to
Corollary 2.1
Let be two positive integers and G a graph of order at least two. Then
3. Outer-independent 2-rainbow domination number
In this section, we focus on outer-independent 2-rainbow domination. An OI2RD-function f on a graph G can be represented by the ordered 4-tuple (or
to refer f) of
, where
,
,
and
. In this representation, its weight is
.
By Theorem 2.1 we immediately obtain
Corollary 3.1
For any graph G of order ,
. Moreover
if and only if G is
or
or
or
.
3.1. Outer-independent 2-rainbow domination versus domination parameters
A function is an outer-independent Roman dominating function (OIRD-function) on G if every vertex
for which
is adjacent to at least one vertex v for which
and
is an independent set. The outer-independent Roman domination number
is the minimum weight of an OIRD-function on G. Outer-independent Roman domination was introduced by Abdollahzadeh Ahangar et al. in [Citation3]. Clearly, if
is a
-function, then the function
is an outer-independent 2-rainbow dominating function on a graph G and so
(2)
(2) Abdollahzadeh Ahangar et al. proved the following bounds on
.
Proposition 3.1
[Citation3]
If G is a connected triangle-free graph of order and maximum degree Δ, then
.
Proposition 3.2
[Citation3]
Let G be a connected graph of order n. If G has girth , then
.
Next results are immediate consequences of Propositions 3.1, 3.2 and inequality (Equation2(2)
(2) ).
Corollary 3.2
If G is a connected triangle-free graph of order and maximum degree Δ, then
. This bound is sharp for all stars
,
.
Corollary 3.3
Let G be a connected graph of order n. If G has girth , then
.
In the following, we provide an upper bound on in terms of
for arbitrary graphs G.
Theorem 3.1
For any graph G, . This bound is sharp for the family
of graphs illustrated in Figure .
Proof.
Let be a
-function and without loss of generality
. Then
is an OIRD-function on G implying that
The notion of outer-independent Italian domination in graphs was introduced in [Citation16]. An outer-independent Italian dominating function (OIID-function) on a graph G is a function such that every vertex
with
has at least two neighbours assigned 1 under f or one neighbour w with
and the set of all vertices assigned 0 under f is independent. The weight of an OIID-function f is the value
. The minimum weight of an OIID-function on a graph G is called the outer-independent Italian domination number
of G. Clearly, if
is a
-function, then the function
is an outer-independent Italian dominating function of G and so
(3)
(3) In [Citation16], the authors proved that
for
and
for
. Using these we obtain the next results.
Proposition 3.3
For ,
.
Proof.
By (Equation3(3)
(3) ), we have
. Let
be a cycle and define
by
,
for
and
otherwise. Clearly f is an OI2RD-function of
and hence
. Thus
.
Proposition 3.4
For ,
.
Proof.
It follows from (Equation3(3)
(3) ) that
. Let
and
be the bipartite sets of
and define
by
,
for
and
otherwise. Clearly f is an OI2RD-function of
and hence
. Therefore
.
Fan et al. [Citation16] proved the following bound on .
Theorem 3.2
For any connected graph G of order with minimum degree δ and maximum degree Δ,
Next result is an immediate consequence of Theorem 3.2 and inequality (Equation3(3)
(3) ).
Corollary 3.4
For any connected graph G of order with minimum degree δ and maximum degree Δ,
This bound is sharp for cycles.
Fan et al. [Citation16] proved the following Nordhaus–Gaddum type result for outer-independent Italian domination number.
Theorem 3.3
For any graph G on n vertices,
As an immediate consequence we have:
Corollary 3.5
For any graph G on n vertices,
The upper bound is sharp for
and the lower bound is sharp for a graph G obtained from
with vertex set
by adding new vertices
and joining
to
and
for
and joining
to all vertices in
.
3.2. Trees
Here we present sharp upper and lower bounds on the outer-independent 2-rainbow domination number of trees. First we show that the outer-independent 2-rainbow domination number and the outer-independent Italian domination number of a tree are equal.
Theorem 3.4
For any tree T, .
Proof.
Consider a -function
on T and let
be the components of the graph
. Let
be the graph whose vertex set is
and two vertices
and
are adjacent if and only if there are vertices
,
and
such that
is a path in T. If
, then we may assume that
. Define
as follows:
for
,
for
,
whenever
and either
or
, where the distance
and
in
, is even, and
otherwise. Clearly g is an OI2RD-function on T with weight
and so
. The result now immediately follows from (Equation3
(3)
(3) ).
Using the results given in [Citation16] and Theorem 3.4, we obtain the following results.
Proposition 3.5
For ,
.
Proposition 3.6
For any tree T of order n, .
Theorem 3.5
For any tree T of order ,
where
is the number of leaves of T. This bound is sharp for stars and paths.
As a consequence of Propositions 2.4 and 3.5 we obtain the following result.
Corollary 3.6
For any connected graph G of order n,
In the sequel we will use the following observation.
Observation 3.1
Let G be a graph.
If u is a strong support vertex of G, then there is a
-function f with
.
If
is a path in G such that
and
, then there is a
-function f with
,
and
.
Proof.
(1) is trivial. To prove (2), let g be a -function. If
, then let f = g. Assume then that
. If
, then since g is a
-function, it follows that
. By the minimality of g, we have
, for otherwise we may assume that
and then the function h defined by
,
and
otherwise, is an OI2RD-function of G with smaller weight than g, a contradiction. Hence we may assume that
. But then the function
defined by
,
,
and
for all
is a
-function with desired property. If
, that is,
, then by the minimality we have
. Now the function f defined above is a
-function with desired property.
Next we present an upper bound on outer-independent 2-rainbow domination number of a tree T in terms of the order and its number of support vertices. For any tree T, let denote the number its support vertices.
Theorem 3.6
If T is a tree of order at least 3, then
This bound is sharp for all paths
.
Proof.
The proof is by induction on . It is easy to verify that the statement is true for
. Hence, let
and assume that every
of order
with
support vertices satisfies
. Let T be a tree of order
If T is a star, then
. Likewise, if T is a double star, then
and so
with equality if and only if
. Henceforth, we assume that
.
If T has a strong support vertex u with , then let
where
. Clearly, there exists a
-function f such that
and f can be extended to an OI2RD-function of T by assigning ∅ to w and this implies that
. Now the result follows by using the induction on
and the facts
and
. Henceforth, we assume that every support vertex of T is adjacent to at most two leaves.
Let be a diametrical path in T such that
is as large as possible and root T at
. We consider the following cases.
Case 1. . If
, then any
-function can be extended to an OI2RD-function of T by assigning
to
and ∅ to the leaves adjacent to
and so
. Using the induction on
and the facts
and
, we obtain
Hence we assume that
. If
, then we have
Let
. We distinguish the followings.
Subcase 1.1. .
If is a support vertex, then any
-function can be extended to an OI2RD-function of T by assigning
to
and ∅ to the leaves adjacent to
and it follows from the induction hypothesis on
and the facts
and
that
Assume next that
is not a support vertex. We consider the following situations.
has a child
with depth one. Let
and let f be a
-function such that
. Without loss of generality, we may assume that
and that
. If
, then the function f can be extended to an OI2RD-function of T by assigning ∅ to
and
to the leaf-neighbour of
, and using the induction hypothesis on
and the facts
and
we obtain
If
, then the function f can be extended to an OI2RD-function of T by assigning a
to
and ∅ to its leaf-neighbours, and using the induction hypothesis on
and the facts
and
we obtain
has a child
with depth two different from
. Suppose
be a path in T such that
. First let
. As in the first paragraph of Case 1, we may assume that
. Let
and let f be a
-function such that
. Without loss of generality, we may assume that
. Then f can be extended to an OI2RD-function of T by assigning
to
and ∅ to
and all leaves of
. Using the induction on
and the facts
and
we obtain
Assume now that
and that all children of
with depth 1 has degree 2. Let
be the number of children of
with depth 0 and
be the number of children of
with depth 1 and let t = 0 if
and t = 1 if
. Suppose
. Clearly, any
-function can be extended to an OI2RD-function of T by assigning
to all leaves at distance 2 from
, ∅ to all children of
, and
to
if
and
to
if
. Using the induction on
and the facts
and
we obtain
Subcase 1.2. .
If , then any
-function can be extended to an OI2RD-function of T by assigning a
to
, a
to
and an ∅ to other vertices in
, and it follows from the induction hypothesis and the facts
and
that
Assume that
. Let
and f be a
-function. By Observation 3.1 (item (2)) we may suppose that
. Then f can be extended to an OI2RD-function of T by assigning a
to
and an ∅ to other vertices in
, and it follows from the induction hypothesis and the facts
and
that
Case 2. .
By the choice of diametrical path, we deduce that every child of with depth one has degree two. Consider the following subcases.
Subcase 2.1. .
First suppose that is a strong support vertex or adjacent to a support vertex except
. Let
and f a
-function. By Observation 3.1, we may assume without loss of generality that
. Now f can be extended to an OI2RD-function of T by assigning a
to
and an ∅ to
. Now we deduce from the induction hypothesis on
and the facts
and
that
Suppose next that
is a support vertex with
. Let
and f a
-function. Since
is a strong support vertex we may assume that
. Now f can be extended to an OI2RD-function of T by assigning a
to
, and as above we have
Subcase 2.2. .
First assume that . Let
. Then any
-function f can be extended to an OI2RD-function of T by assigning a
to
, a
to
and an ∅ to
, and it follows from the induction hypothesis and the facts
and
that
Assume next that
. Let
and f a
-function. Observation 3.1 (item (2)), we may assume that
. Now f can be extended to an OI2RD-function of T by assigning a
to
and an ∅ to
, and we deduce from the induction hypothesis and the facts
and
that
and this completes the proof.
Next result in an immediate consequence of Theorems 3.4 and 3.6 since the number of support vertices of a tree on vertices is at most
.
Corollary 3.7
For any tree T of order ,
A constructive instruction of trees attaining the bound given in Corollary 3.7 is given in [Citation16].
We close this section by establishing a lower bound on the outer-independent 2-rainbow domination number of a tree in terms of the order and the total outer-independent domination number. Recall that a set S of vertices of a graph G is a total outer-independent dominating set if every vertex from has a neighbour in S and the complement of S is an independent set. The total outer-independent domination number
of G is the smallest possible cardinality of any total outer-independent dominating set of G. The total outer-independent domination was introduced in [Citation17, Citation18]. It was observed in [Citation17] that
Theorem 3.7
For any nontrivial tree T,
Furthermore, this bound is sharp for paths
.
Proof.
The proof is by induction on the order . If
or
, then
and the bound is sharp. Let
and assume that for any tree
of order
,
. Let T be a tree of order
and
be a
-function. Since for stars and double stars T we have
, so the statement holds. Hence, we assume that T has diameter at least four. If
, then by the fact that
we have
. Therefore we assume that
. First suppose that T has a support vertex x which is adjacent to two or more leaves. Let u, v be two leaves adjacent to x and
be the tree obtained from T by removing u. Obviously,
. Also by Observation 3.1, we may assume that
and so all leaves adjacent to x belong to
. Hence, the function f, restricted to
, is a OI2RD-function on
, implying that
. Applying the inductive hypothesis to
, we get
, as desired. Therefore, we may assume that every support vertex of T is adjacent to exactly one leaf. Suppose
, and let
be a diametrical path of T. Root T at
. Thus,
is a leaf of T and
.
By item (2) of Observation 3.1, there is a -function g such that
,
and
. Consider the following cases.
Case 1. .
Let . Since
is a support vertex or is adjacent to a support vertex of degree two in
, any
-set containing no leaf, contains
and so it can be extended to an OITD-set of T by adding
implying that
. On the other hand, the function g restricted to
is an OI2RD-function of
of weight at most
and we conclude from the induction hypothesis that
Case 2.
.
If or
, then let
. Clearly the function g restricted to
is an OI2RD-function of
of weight at most
and so
. On the other hand, any
-set can be extended to an OITD-set of T by adding
yielding
. It follows from the induction hypothesis that
Assume that
and that
. Since
is independent and
, we may assume without loss of generality that
for each
. First let
. Assume
and let
be the components of
containing
. Define
by
if
and
, and
otherwise. Clearly, h is a
-function and we are in above situation and so we have
.
Now let . Assume that
. Then the function g restricted to
is an OI2RD-function of
, implying that
. On the other hand, any
-set can be extended to an total outer-independent set of T by adding
, yielding
. We deduce from the induction hypothesis on
that
This completes the proof.
We conclude this paper with an open problem.
Problem. Prove or disprove: for any non-trivial connected graph G of order n, .
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Qiong Kang http://orcid.org/0000-0001-8123-9184
Vladimir Samodivkin http://orcid.org/0000-0001-7934-5789
Marzieh Soroudi http://orcid.org/0000-0002-6677-6835
Additional information
Funding
References
- West DB. Introduction to graph theory. 2nd ed. Upper Saddle River: Prentice-Hall, Inc.; 2000.
- Gallai T. Uber extreme Punkt- und Kantenmengen. Ann Univ Sci Budapest Eotvos Sect Math. 1959;2:133–138.
- Ahangar HA, Chellali M, Samodivkin V. Outer independent Roman dominating functions in graphs. Int J Comput Math. 2017;94:2547–2557. doi: 10.1080/00207160.2017.1301437
- Krzywkowski M, Venkatakrishnan YB. Bipartite theory of graphs: outer-independent domination. Natl Acad Sci Lett. 2015;38:169–172. doi: 10.1007/s40009-014-0315-7
- Krzywkowski M. An upper bound on the 2-outer-independent domination number of a tree. C R Math. 2011;349:1123–1125. doi: 10.1016/j.crma.2011.10.005
- Li Z, Shao Z, Lang F, et al. Computational complexity of outer-independent total and total Roman domination numbers in trees. IEEE Access. 2018;6:35544–35550. doi: 10.1109/ACCESS.2018.2851972
- Rad NJ, Krzywkowski M. 2-Outer-independent domination in graphs. Natl Acad Sci Lett. 2015;38:263–269. doi: 10.1007/s40009-015-0389-x
- Brešar B, Henning MA, Rall DF. Rainbow domination in graphs. Taiwanese J Math. 2008;12:201–213. doi: 10.11650/twjm/1500602498
- Amjadi J, Dehgardi N, Furuya M, et al. A sufficient condition for large rainbow domination number. Int J Comput Math Comput Syst Theory. 2017;2:53–65. doi: 10.1080/23799927.2017.1330282
- Brešar B, Kraner Šumenjak T. On the 2-rainbow domination in graphs. Discrete Appl Math. 2007;155:2394–2400. doi: 10.1016/j.dam.2007.07.018
- Chang GJ, Wu J, Zhu X. Rainbow domination on trees. Discrete Appl Math. 2010;158:8–12. doi: 10.1016/j.dam.2009.08.010
- Chellali M, Haynes TW, Hedetniemi ST. Bounds on weak roman and 2-rainbow domination numbers. Discrete Appl Math. 2014;178:27–32. doi: 10.1016/j.dam.2014.06.016
- Meierling D, Sheikholeslami SM, Volkmann L. Nordhaus–Gaddum bounds on the k-rainbow domatic number of a graph. Appl Math Lett. 2011;24:1758–1761. doi: 10.1016/j.aml.2011.04.046
- Sheikholeslami SM, Volkmann L. The k-rainbow domatic number of a graph. Discuss Math Graph Theory. 2012;32:129–140. doi: 10.7151/dmgt.1591
- Xu G. 2-rainbow domination in generalized Petersen graphs P(n,3). Discrete Appl Math. 2009;157:2570–2573. doi: 10.1016/j.dam.2009.03.016
- Fan W, Ye A, Miao F, et al. Outer-independent Italian domination in graphs. IEEE Access. 2019;7:22756–22762. doi: 10.1109/ACCESS.2019.2899875
- Krzywkowski M. Total outer-independent domination in graphs, Manuscript.
- Krzywkowski M. A lower bound on the total outer-independent domination number of a tree. C R Math. 2011;349:7–9. doi: 10.1016/j.crma.2010.11.021