RESEARCH PAPER
Graphs with Large Total 2-Rainbow Domination Number
H. Abdollahzadeh Ahangar1•M. Khaibari1•N. Jafari Rad2• S. M. Sheikholeslami3
Received: 19 April 2016 / Accepted: 13 December 2017 ÓShiraz University 2018
Abstract
LetG¼ ðV;EÞbe a simple graph with no isolated vertex. A 2-rainbow dominating function(2RDF) ofGis a function ffrom the vertex setV(G) to the set of all subsets of the setf1;2g such that for any vertexv2VðGÞwithfðvÞ ¼ ;the conditionS
u2NðvÞfðuÞ ¼ f1;2gis fulfilled, whereN(v) is the open neighborhood ofv. A 2-rainbow dominating functionfis
called atotal 2-rainbow dominating function (T2RDF) if the subgraph of Ginduced by fv2VðGÞ jfðvÞ 6¼ ;g has no isolated vertex. The weight of a T2RDFfis defined as wðfÞ ¼P
v2VðGÞjfðvÞj. Thetotal2-rainbow domination number,
ctr2ðGÞ, is the minimum weight of a total 2-rainbow dominating function onG. In this paper, we characterize all graphs Gwhose total 2-rainbow domination number is equal to their order minus one.
Keywords 2-rainbow dominating function2-rainbow domination numberTotal 2-rainbow dominating function Total 2-rainbow domination number
Mathematics Subject Classification 05C69
1 Introduction
For notation and graph theory terminology, we in general follow Haynes et al. (1998a, b) and Henning and Yeo (2013). In this paper, we continue the study of rainbow domination number in graphs. Specifically, let G be a simple graph with vertex setV¼VðGÞ, edge setE¼EðGÞ and with no isolated vertex. The order |V| ofGis denoted byn and size |E| ofG is denoted bym. For every vertex v2V, the open neighborhood NGðvÞ ¼NðvÞ is the set fu2Vjuv2Eg and the closed neighborhoodof vis the setNG½v ¼N½v ¼NðvÞ [ fvg. Thedegreeof a vertexv2
V is degGðvÞ ¼degðvÞ ¼ jVðvÞj. The minimum and the maximum degree of a graph G are denoted by d¼dðGÞ andD¼DðGÞ, respectively. The open neighborhoodof a set SV is the set NðSÞ ¼ [v2SNðvÞ, and the closed neighborhood of Sis the set N½S ¼NðSÞ [S. Aleafis a vertex of degree one, asupport vertexis a vertex adjacent to a leaf, aweak support vertexis a support vertex adjacent to exactly one leaf, and astrong support vertexis a support vertex adjacent to at least two leaves. We denote the set of leaves and support vertices of G by L(G) and S(G), respectively. We also denote by Lv the set of all leaves adjacent to a support vertex v. We write Kn for the com- plete graphof ordern,Cnfor acycleof ordernandPnfor a pathof ordern. ThediameterofG, denoted by diamðGÞ, is the maximum value among minimum distances between all pairs of vertices ofG. ThegirthofG, denoted byg(G), is the minimum length of a cycle in G. The corona graph HK1 of a graph H is a graph obtained from H by attaching a leaf to every vertex of H. For a subset S of vertices ofG, we denote byG[S] the subgraph induced by S.
A subsetSof vertices of a graphGis adominating setof G if every vertex in VðGÞ S has a neighbor in S. The domination number cðGÞis the minimum cardinality of a dominating set ofG. A dominating setSin a graph with no
& H. Abdollahzadeh Ahangar
[email protected] N. Jafari Rad [email protected] S. M. Sheikholeslami
1 Department of Mathematics, Babol Noshirvani University of Technology, Babol, Islamic Republic of Iran
2 Department of Mathematics, Shahrood University of Technology, Shahrood, Islamic Republic of Iran
3 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Islamic Republic of Iran
https://doi.org/10.1007/s40995-017-0465-9(0123456789().,-volV)(0123456789().,-volV)
isolated vertex is called a total dominating set of G if G[S] has no isolated vertex. Thetotal domination number ctðGÞis the minimum cardinality of a total dominating set ofG. The literature on the subject of domination parame- ters in graphs, through 1998, has been surveyed in Haynes et al. (1998a, b), and the subject of total domination in graphs, through 2013, has been surveyed in Henning and Yeo (2013). Recently, Liu and Chang (2013) introduced the concept of total Roman domination in graphs albeit in a more general setting. And very recently has been studied by Abdollahzadeh Ahangar et al. (2016,2017).
A 2-rainbow dominating function(2RDF) of a graphG is a function f from the vertex set V(G) to the set of all subsets of the setf1;2gsuch that for any vertexv2VðGÞ with fðvÞ ¼ ; the condition [u2NðvÞfðuÞ ¼ f1;2g is ful- filled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is defined as wðfÞ ¼P
v2VðGÞjfðvÞj.
The minimum weight of a 2-rainbow dominating function is called the 2-rainbow domination number ofG, denoted by cr2ðGÞ. The concept of 2-rainbow domination was introduced by Bresˇar et al. (2008), and has been studied by several authors, for example Chang et al. (2010), Chellali and Jafari Rad (2013), Chunling et al. (2009), Bresˇar and Sumenjak (2007), Dehgardi et al. (2015), Falahat et al.
(2014), Meierling et al. (2011), Sheikholeslami and Volk- mann (2012), Wu and Jafari Rad (2013), Wu and Xing (2010) and Xu (2009).
If f is a 2-rainbow dominating function in a graph G, then clearly fv2VðGÞ jfðvÞ 6¼ ;gis a dominating set of G. Abdollahzadeh Ahangar et al. (2018a) considered a variant of 2-rainbow dominating functionsfsuch thatfv2 VðGÞ jfðvÞ 6¼ ;gis a total dominating set ofG. They thus introduced the concept of total 2-rainbow domination in graphs. A 2-rainbow dominating function fis called atotal 2-rainbow dominating function, or just T2RDF if the sub- graph ofGinduced byfv2VðGÞ jfðvÞ 6¼ ;ghas no iso- lated vertex. The total 2-rainbow domination number, ctr2ðGÞ, is the minimum weight of a total 2-rainbow dominating function onG. Clearly,ctr2ðGÞis well defined for any graphG with no isolated vertex, since assigning f1g to every vertex yields a total 2-rainbow dominating function and hence
ctr2ðGÞ jVðGÞj: ð1Þ
It is shown in Abdollahzadeh Ahangar et al. (2018b) that the decision version of the total 2-rainbow domination problem is NP-complete.
A 2-rainbow dominating functionfcan be represented by the ordered partitionf ¼ ðV0f;V1f;V2f;V12f Þ, whereV0f ¼ fvj fðvÞ ¼ ;g, V1f ¼ fvjfðvÞ ¼ f1gg, V2f ¼ fvjfðvÞ ¼ f2gg, andV12f ¼ fvjfðvÞ ¼ f1;2gg. Thus,f is a total 2-rainbow
dominating function if the subgraph ofGinduced byV1f [ V2f[V12f has no isolated vertex.
Note that if G1;G2;. . .;Gs are the components of G, thenctr2ðGÞ ¼Ps
i¼1ctr2ðGiÞ. Hence, it would be sufficient to consider only connected graphs in the study of total 2-rainbow domination.
Our purpose in this paper is to characterize all the connected graphs Gwhose total 2-rainbow domination is equal to their order minus one.
We make use of the following observations in this paper.
Observation 1 IfGis a connected graph of ordern4, and G has a support vertex x with jLxj 2, then ctr2ðGÞ n1.
Observation 2 If G is a connected graph of order n, different from K1;3, and G has a support vertex x with jLxj 3, thenctr2ðGÞ n2.
Observation 3 IfGis a connected graph of ordernandG has two support verticesxandywithjLxj 2 andjLyj 2, thenctr2ðGÞ n2.
Observation 4 IfGis a connected graph of ordernwith diamðGÞ ¼2, then ctr2ðGÞ 2dðGÞ þ1.
Proof Let x be a vertex of minimum degree dðGÞ and definef :VðGÞ ! Pðf1;2gÞbyfðxÞ ¼ f1g; fðuÞ ¼ f1;2g for u2NðxÞ and fðyÞ ¼ ; if y62N½x. Clearly, f is a T2RDF on G of weight 2dðGÞ þ1 and so
ctr2ðGÞ 2dðGÞ þ1. h
2 Graphs with Large Total 2-Rainbow Domination Number
Assume D ¼ fHK1jHis a connected graphg. In Abdollahzadeh Ahangar et al. (2018a), the authors char- acterized all graphs Gfor whichctr2ðGÞ ¼n as follows:
Theorem 5 (Abdollahzadeh Ahangar et al. 2018a) Let G be a connected graph of order n. Then,ctr2ðGÞ ¼n if and only if G2 D [ fP3g.
Here, we characterize all connected graphs Gof order n3 withctr2ðGÞ ¼n1. We start with some lemmas.
Lemma 6 Let G be a connected graph of order n with dðGÞ 2.Then,ctr2ðGÞ ¼n1if and only if G¼C3,C4 or C5.
Proof One side is clear. Let ctr2ðGÞ ¼n1. If diamðGÞ 3 andw1w2. . .wkis a diametral path inG, then let u1 2Nðw1Þ fw2g, v12NðwkÞ fwk1g, and define the function f :VðGÞ ! Pðf1;2gÞ by fðu1Þ ¼fðv1Þ ¼
f1g; fðw1Þ ¼fðwkÞ ¼ ; and fðwÞ ¼ f2g otherwise. It is easy to verify thatf is a T2RDF onGof weight n2, a contradiction. Thus, diamðGÞ 2. If diamðGÞ ¼1, then Gis the complete graph of order n and we deduce from ctr2ðKnÞ ¼2 that G¼K3 by the assumption. Henceforth, we assume diamðGÞ ¼2. By Observation 4, we have dðGÞ n22 . Let x be a vertex of minimum degree dðGÞ, NðxÞ ¼ fx1;x2. . .;xkg and X¼VðGÞ N½x. Since diamðGÞ ¼2 and dðGÞ n22 , we have 1 jXj n2. AssumeX¼ fz1;z2;. . .;zjXjg.
IfjXj ¼1 and dðGÞ 3, then the function f :VðGÞ ! Pðf1;2gÞdefined byfðxÞ ¼fðz1Þ ¼ ;;fðx1Þ ¼ f1g;fðvÞ ¼ f2gforv2NðxÞ fx1g, is a TR2DF ofGof weightn2, a contradiction. If jXj ¼1 and dðGÞ ¼2, then clearly G¼C4. LetjXj 2.
IfDðG½XÞ 2 andy2Xhas degree at least two inG[X], then the function f :VðGÞ ! Pðf1;2gÞ defined by fðyÞ ¼ f1g;fðvÞ ¼ ; forv2NðyÞ \X,fðuÞ ¼ f2g otherwise, is a TR2DF ofGof weight at mostn2, a contradiction. Assume thatDðG½XÞ 1. Then, all components ofG[X] areK1orK2. It follows that each vertex inXhas at leastdðGÞ 1 neighbors in N(x) and every two vertices in X have at least dðGÞ 2 common neighbors inN(x). IfdðGÞ 4, then the functionf : VðGÞ ! Pðf1;2gÞ defined byfðx1Þ ¼fðx2Þ ¼ f1g;fðuÞ ¼
;foru2X andfðvÞ ¼ f2gotherwise, is a TR2DF ofGof weight at mostn2, a contradiction. IfG[X] has two isolated vertices, say z1;z2, then Nðz1Þ ¼Nðz2Þ ¼NðxÞ and the function f :VðGÞ ! Pðf1;2gÞ defined by fðz1Þ ¼fðz2Þ ¼
;;fðx1Þ ¼ f1gandfðvÞ ¼ f2gotherwise, is a TR2DF ofGof weight n2, a contradiction. Henceforth, we assume that dðGÞ 3 andG[X] has at most one isolated vertex that implies G[X] has aK2component, sayz1z2.
First letdðGÞ ¼3. Then, we have n8 andjXj 4. If jXj ¼4, then the functionf :VðGÞ ! Pðf1;2gÞdefined by fðx1Þ ¼fðx2Þ ¼ f1;2g;fðxÞ ¼fðx3Þ ¼ f1g and fðuÞ ¼ ; foru2X, is a TR2DF ofGof weightn2, a contradiction.
We may assume, therefore, thatjXj 3. SincedðGÞ ¼3, we may assume thatx12Nðz1Þ \Nðz2Þ. Now, it is easy to verify that the functionf :VðGÞ ! Pðf1;2gÞdefined by fðx1Þ ¼ f1g; fðxÞ ¼fðx2Þ ¼fðx3Þ ¼ f2gandfðuÞ ¼ ;foru2X, is a TR2DF ofGof weightn2 which leads to a contradiction.
Now letdðGÞ ¼2 and let without loss of generality that degðx1Þ degðx2Þ. Then, we have n6 and jXj 3. If n¼6, then the functionf :VðGÞ ! Pðf1;2gÞdefined by
fðx1Þ ¼ f1;2g;fðxÞ ¼ f1g; fðx2Þ ¼ f2gandfðuÞ
¼ ;otherwise;
whenDðG½XÞ ¼0 orDðG½XÞ ¼1 and degðx1Þ degðx2Þ, and by
fðx1Þ ¼ f1g;fðxÞ ¼fðzÞ ¼ ;for exactly one z2Nðx1Þ
\Z; andfðuÞ ¼ f2gotherwise;
ifDðG½XÞ ¼1 and degðx1Þ ¼degðx2Þ, is a TR2DF ofGof weight 4, a contradiction. Thus,n¼5 and sojXj ¼2. It is easy to check that G¼C5 in this case and the proof is
complete. h
Assume D1 is the family of graphs consisting of all graphsGsuch thatGis obtained from a graph inD [ fP3g by adding a pendant edge to precisely one support vertex.
Lemma 7 Let G be a graph such that each vertex of G is a leaf or a support vertex. Then, ctr2ðGÞ ¼n1 if and only if G2 D1.
Proof Let ctr2ðGÞ ¼n1. Since each vertex of G is a leaf or a support vertex, it follows from Theorem 5 that G has a strong support vertex. If G¼K1;3, then clearly G2 D1. LetG6¼K1;3. It follows from Observations2and 3thatGhas exactly one support vertex which is adjacent to exactly two leaves. Thus,G2 D1.
Conversely, let G2 D1. Then, G is obtained from a graph inD [ fP3g, by adding a pendant edge to precisely one support vertex of degree at least two. By Theorem5, we have ctr2ðGÞ n1. Now, we show that ctr2ðGÞ n1. Letfbe actr2ðGÞ-function. IfGis obtained from P3, then G¼K1;3 and clearly ctr2ðGÞ ¼3. Assume Gis obtained fromHK1 for some connected graphHof order at least two. Let VðHÞ ¼ fu1;u2. . .;umg, vi be the leaf adjacent to ui for each i, and w be the new pendant vertex joined to um. Since f is a TR2DF of G, we have jfðuiÞj þ jfðviÞj 2 for each i and this implies that
ctr2ðGÞ 2m¼n1 as desired. h
Regarding Lemmas 6 and 7, it is enough to consider graphsGsuch thatdðGÞ ¼1 andSðGÞ [LðGÞ$VðGÞ.
Lemma 8 Let G be a connected graph of order n with dðGÞ ¼1and SðGÞ [LðGÞ$VðGÞ.Ifctr2ðGÞ ¼n1,then the induced subgraph G½VðGÞ ðSðGÞ [LðGÞÞ is iso- morphic to K1;K2;P3;C3; or C4.
Proof First, we show that G ðSðGÞ [LðGÞÞ is a con- nected graph. Assume, to the contrary, that B1 andB2 are two components of G ðSðGÞ [LðGÞÞ and xi2Bi for i¼1;2. Since xi62SðGÞ [LðGÞ, xi has two neighbors x1i;x2i each of degree at least two, for i¼1;2. Then, the functionf :VðGÞ ! Pðf1;2gÞdefined byfðx1Þ ¼fðx2Þ ¼
;, fðx11Þ ¼fðx12Þ ¼ f2g and fðxÞ ¼ f1g otherwise, is a TR2DF of G of weight at most n2, a contradiction.
Hence,Gis connected. Consider two cases.
Case 1G½VðGÞ ðSðGÞ [LðGÞÞcontains a cycle.
Assume C¼ ðx1x2. . .xkÞ is a cycle of G½VðGÞ ðSðGÞ [LðGÞÞ. If k6, then the function f : VðGÞ ! Pðf1;2gÞ defined by fðx1Þ ¼fðx2Þ ¼ f1g;fðx3Þ ¼fðxkÞ ¼ ; and fðxÞ ¼ f2g otherwise, is a
TR2DF of G of weight n2, a contradiction. If k¼5, then since G is connected and VðGÞ 6¼SðGÞ [LðGÞ, we may assume thatx1 has a neighbor woutsideV(C). Then, the function f :VðGÞ ! Pðf1;2gÞ defined by fðx1Þ ¼ fðwÞ ¼ f1g;fðx2Þ ¼fðx5Þ ¼ ; and fðxÞ ¼ f2g otherwise, is a TR2DF ofGof weight at mostn2, a contradiction.
Thus,k4.
First let k¼4. If C has a chord, say x1x3, then the function f ¼ ðfx2;x4g;VðGÞ fx1;x2;x4g;fx1g;;Þ is a TR2DF of G of weight n2, a contradiction. SoC has no chord. IfG½VðGÞ ðSðGÞ [LðGÞÞ 6¼C, then we may assumex1 has a neighbor w outside SðGÞ [LðGÞand the functionf :VðGÞ ! Pðf1;2gÞ defined byfðx4Þ ¼fðwÞ ¼
;, fðx1Þ ¼ f2g andfðxÞ ¼ f1g otherwise, is a TR2DF of G of weight n2, a contradiction. Hence, G½VðGÞ ðSðGÞ [LðGÞÞ ffiC4 in this case.
Now, letk¼3. IfG½VðGÞ ðSðGÞ [LðGÞÞ 6¼C, then we may assumex1 has a neighborwoutside SðGÞ [LðGÞ and the functionf ¼ ðfw;x3g;VðGÞ fx1;w;x3g;fx1g;;Þ is a TR2DF of Gof weight n2, a contradiction. Thus, G½VðGÞ ðSðGÞ [LðGÞÞ ffiC3 in this case.
Case 2G½VðGÞ ðSðGÞ [LðGÞÞis a tree.
IfDðG½VðGÞ ðSðGÞ [LðGÞÞÞ 3, then letxbe a vertex of maximum degree inG½VðGÞ ðSðGÞ [LðGÞÞ and let y1;y2;y3 be the neighbors of x in G½VðGÞ ðSðGÞ [LðGÞÞ. Then, the functionf :VðGÞ ! Pðf1;2gÞ defined by fðy2Þ ¼fðy3Þ ¼ ;, fðxÞ ¼fðy1Þ ¼ f2gandfðxÞ ¼ f1gotherwise, is a TR2DF ofGof weight n2, a contradiction. Thus, DðG½VðGÞ ðSðGÞ [ LðGÞÞÞ 2 and so G½VðGÞ ðSðGÞ [LðGÞÞ is a path
Pm¼x1x2. . .xm. If m4, then the function f :VðGÞ !
Pðf1;2gÞ defined by fðx1Þ ¼fðxmÞ ¼ ;, fðxiÞ ¼ f2g for 2im1, and fðxÞ ¼ f1g otherwise, is a TR2DF of G of weight n2, a contradiction. Thus, m3 and the
proof is complete. h
Recall that D ¼ fHK1jHis a connected graphg.
Now, we introduce the following families of graphs.
1. D2 is the family of graphs consisting of all graphs G such that G is obtained from r2 graphs
G1;G2;. . .;Gr2 D, by adding a new vertex, called
head, and joining it to a support vertex ofGi, for each i¼1;2;. . .;r.
2. D3 is the family of graphs consisting of all graphs G such that G is obtained from r1 graphs
G1;G2;. . .;Gr2 D and P3, by adding a new vertex,
called head, and joining it to the support vertex ofP3
and a support vertex ofGi, for eachi¼1;2;. . .;r.
3. D4 is the family of graphs consisting of all graphs G such that G is obtained from r1 graphs G1;G2;. . .;Gr2 D, whereGr¼K2, by adding a path
x1x2 and joiningx1to a support vertex ofGi, for each i¼1;2;. . .;r1 andx2 to a leaf ofGr.
4. D5 is the family of graphs consisting of all graphs G such that G is obtained from two graphs G1;G22 D2, by adding a new vertex and joining it to the head ofDi for i¼1;2.
5. D6 is the family of graphs consisting of all graphs G such that G is obtained from r2 graphs
G1;G2;. . .;Gr2 D, by adding a path xyz, joining
xto a support vertex ofGi, for eachi¼1;2;. . .;r1 and joiningzto a support vertex ofGr.
6. D7be a family of graphs consisting of all graphsGsuch thatGis obtained from a cycleðx1x2x3Þby joiningx1to a support vertex of finitely many graphsG11;G12;. . .;G1r
1 2
D ðr11Þandx2 to a support vertex of finitely many graphsG21;G22;. . .;G2r
22 D(possibly no graphs).
7. D8 be a family of graphs consisting of all graphs Gsuch thatGis obtained from a cycleC4¼ ðx1x2x3x4Þ by joining x1 to a support vertex of finitely many graphsG1;G2;. . .;Gr2 D ðr1Þ.
Lemma 9 Let G be a graph of order n. If G2 [8i¼1Di, thenctr2ðGÞ ¼n1.
Proof LetG2 [8i¼1Di. IfG2 D1, thenctr2ðTÞ ¼n1 by Lemma 7. Suppose T2 [8i¼2Di. By Theorem 5, we have ctr2ðGÞ n1. Now, we show thatctr2ðGÞ n1. Sup- pose fis actr2ðGÞ-function.
LetG2 D2. Then,Gis obtained fromr2 graphsH1
K1;H2K1;. . .;HrK1 for each i, by adding a new
vertex and joining it to a support vertex of Hi for i¼1;2;. . .;r. Let VðHiÞ ¼ fui1;ui2. . .;uim
ig and vij be the leaf adjacent to uij for each i and j. Clearly, jfðuijÞj þ jfðvijÞj 2 i, j and so ctr2ðGÞ ¼xðfÞ Pr
i¼12jVðHiÞj ¼ n1 as desired.
LetG2 D3. Then,Gis obtained fromr1 graphsH1
K1;H2K1;. . .;HrK1 for each i, by adding a new
vertex w and joining it to the support vertex of a path P3 ¼z1z2z3 and to a vertex ofHi, for each i¼1;2;. . .;r.
LetVðHiÞ ¼ fui1;ui2;. . .;uim
igandvijbe the leaf adjacent to uij for each i andj. Clearly, jfðwÞj þ jfðz1Þj þ jfðz2Þj þ jfðz3Þj 3 and jfðuijÞj þ jfðvijÞj 2 for each i,j and so ctr2ðGÞ ¼xðfÞ 3þPr
i¼12jVðHiÞj ¼n1 as desired.
LetG2 D4. Then,Gis obtained fromr2 graphsH1 K1;H2K1;. . .;Hr1K1 andGr ¼K2 ¼z1z2by adding a path x1x2, joining x2 toz2 andx1 to a vertex of Hi, for eachi¼1;2;. . .;r1. LetVðHiÞ ¼ fui1;ui2;. . .;uim
ig and
vij be the leaf adjacent to uij for each i andj. Clearly, jfðx1Þj þ jfðx2Þj þ jfðz1Þj þ jfðz2Þj 3 and as above we have ctr2ðTÞ ¼xðfÞ Pr
i¼12jVðHiÞj ¼n1 as desired.
Let G2 D5. Then, G is obtained from graphs H11K1;H21K1;. . .;Hm11K1;H21 K1;H22K1;. . .;
Hm22K1, withm1;m22, by adding a pathP3 ¼x1x2x3, and joining xi to a support vertex of Hji, for each j and i¼1;3. To totally rainbowly dominate x2 we must have jfðx1Þj þ jfðx2Þj þ jfðx3Þj 2 and as above case we obtain
ctr2ðGÞ ¼xðfÞ 3þX
2
i¼1
X
mi
j¼1
X
x2VðHjiK1Þ
jfðxÞj 0
@
1 A
¼3þX2
i¼1
Xmi
j¼1
jVðHijK1Þj n1;
as desired. The proofs for the casesG2 D6,G2 D7 and
G2 D8 are similar. h
Lemma 10 Let G be a connected graph of order n with dðGÞ ¼1 and LðGÞ [SðGÞ$VðGÞ. Then, ctr2ðGÞ ¼n1 if and only if G2 [8i¼2Di.
Proof Sufficiency is true by Lemma9. It is enough to prove necessity. Letctr2ðGÞ ¼n1. Clearly,n4. Ifn¼4, then G¼K1;32 D1. Assume n5. If VðGÞ ¼LðGÞ [SðGÞ, then G2 D1 by Lemma 7. Let LðGÞ [SðGÞ$VðGÞ. By Lemma 8, G½VðGÞ ðSðGÞ [LðGÞÞ is isomorphic to K1;P2;P3;C3; orC4. We consider the following cases.
Case 1G½VðGÞ ðSðGÞ [LðGÞÞffiK1¼w.
Let G1;G2;. . .;Gk be the components of Gw. We
deduce from G½VðGÞ ðSðGÞ [LðGÞÞ ¼w that k2, jVðGiÞj 2 for each i, and w is adjacent to a support vertex wi of Gi for each i. It follows that Gi has a ctr2- function gi such that 12giðwiÞ for each i. If ctr2ðGiÞ jVðGiÞj 1 for some i, say i¼1, then the function f :VðGÞ ! Pðf1;2gÞ defined by fðwÞ ¼
;;fðuÞ ¼g1ðuÞfor u2VðG1Þ, and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight n2, a contradiction. Thus, ctr2ðGiÞ ¼ jVðGiÞjfor eachi. This implies thatGi2 D [P3
for 1ik. If Gi¼P3 ¼y1wiy3 and Gj¼P3¼y01wjy03 for some i6¼j, then the function f :VðGÞ ! Pðf1;2gÞ defined by fðy1Þ ¼fðy3Þ ¼fðy01Þ ¼fðy03Þ ¼ ;;fðwiÞ ¼ fðwjÞ ¼ f1;2g;fðwÞ ¼1, and fðuÞ ¼ f1g otherwise, is a TR2DF ofTof weight at mostn2, a contradiction. Thus at most one ofGi’s isP3. This implies thatG2 D2[ D3. Case 2G½VðGÞ ðSðGÞ [LðGÞÞffiP2¼x1x2.
Let Gi1;Gi2;. . .;Gim
i be the components of G fx1;x2g adjacent toxi fori¼1;2. Sincex1 andx2 are not support vertices,jVðGijÞj 2 fori¼1;2 and eachj. Ifm1;m22, then the functionf :VðGÞ ! Pðf1;2gÞdefined byfðx1Þ ¼ fðx2Þ ¼ ;;fðuÞ ¼ f1g for u2VðG11Þ [VðG21Þ, and fðuÞ ¼
f2g otherwise, is a TR2DF of T of weight n2, a contradiction. Assume that m2¼1. Since x1;x2 are the only vertices ofGwhich are not leaf or support vertex,x1is adjacent to a support vertex ofG1j for each 1jm1 and x2 is adjacent to a support vertex w2 of G21. If ctr2ðG21Þ jVðG21Þj 1, then let g be a ctr2ðG21Þ-function so that 12gðw2Þ and define f :VðGÞ ! Pðf1;2gÞ by fðx2Þ ¼ ;;fðuÞ ¼gðuÞ for u2VðG21Þ, and fðuÞ ¼ f2g otherwise. Clearly, f is a TR2DF of T of weight at most n2, a contradiction. Thus,ctr2ðG21Þ ¼ jVðG21Þj. It is easy to verify thatG216¼P3 and soG212 D.
IfG21 ¼w2w02, then as above we can see thatctr2ðG1jÞ ¼ jVðG1jÞjfor eachj, and soG2 D4. LetjVðG21Þj 4. Then, G21has actr2-functiong2so thatg2ðw2Þ ¼ f1;2g. Ifm12, then the functionf :VðGÞ ! Pðf1;2gÞdefined byfðx1Þ ¼ fðx2Þ ¼ ;;fðuÞ ¼g2ðuÞforu2VðG21Þ,fðuÞ ¼ f1gforu2 VðG11Þ and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight n2, a contradiction. Therefore, m1 ¼1. As above, we can see that ctr2ðG11Þ ¼ jVðG11Þj and G116¼P3. Let x1 be adjacent to the support vertex w1 of G11. If jVðG11Þj 4, then let g1 be a ctr2ðG11Þ-function such that g1ðw1Þ ¼ f1;2g and define the function f :VðGÞ ! Pðf1;2gÞ by fðx1Þ ¼fðx2Þ ¼ ;;fðuÞ ¼g1ðuÞ for u2VðG11Þ, and fðuÞ ¼g2ðuÞ for u2VðG21Þ. Then, f is a TR2DF of T of weight at most n2 which is a contradiction. Thus,jVðG11Þj ¼2 that impliesG2 D4. Case 3G½VðGÞ ðSðGÞ [LðGÞÞffiP3 ¼x1x2x3.
If degðx2Þ 3, then the function f :VðGÞ ! Pðf1;2gÞ defined byfðx1Þ ¼fðx3Þ ¼ ;;fðx2Þ ¼ f1g, andfðuÞ ¼ f2g otherwise, is a TR2DF ofT of weightn2, a contradic- tion. Hence, degðx2Þ ¼2. Assume Gi1;Gi2;. . .;Gim
i are the components of G fx1;x2;x3g adjacent toxi for i¼1;3.
Since x1 and x3 are not support vertices, jVðGijÞj 2 for each i andj¼1;3. If ctr2ðG1jÞ jVðG1jÞj 1 for some j, then let g be a ctr2ðG1jÞ-function and define f :VðGÞ ! Pðf1;2gÞ by fðx3Þ ¼ ;;fðx2Þ ¼ f1g;fðuÞ ¼gðuÞ for u2VðG1jÞ, and fðuÞ ¼ f2g otherwise. Clearly, f is a TR2DF ofTof weight at mostn2, a contradiction. Thus, ctr2ðG1jÞ ¼ jVðG1jÞj for each 1ijm1. If G1j ¼P3¼ y1y2y3 for some j, then define f :VðGÞ ! Pðf1;2gÞ by fðy1Þ ¼fðy3Þ ¼fðx2Þ ¼ ;;fðy2Þ ¼ f1;2g;fðx1Þ ¼ f1gand fðuÞ ¼ f2g otherwise, when x1y22EðGÞ, and by fðy1Þ ¼ fðx3Þ ¼ ;;fðx1Þ ¼ f1g and fðuÞ ¼ f2g otherwise, when x1y12EðGÞ. Clearly,fis a TR2DF ofTof weight at most n2, a contradiction. Thus, G1j 6¼P3 and so G1j 2 D for each j. Similarly, G2j 2 D for each 1jm3. If jVðG1jÞj 4 for somej, andx1 is adjacent to a leaf ofG1j such asv, then the functionf :VðGÞ ! Pðf1;2gÞdefined
byfðx3Þ ¼fðvÞ ¼ ;;fðx1Þ ¼fðx2Þ ¼ f1g, and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight at most n2, a contradiction. Thus,x1is adjacent to a support vertex ofG1j for 1jm1. Similarly,x3 is adjacent to a support vertex ofG2j for each 1jm3. Thus,G2 D6ifm1;m22 and G2 D7 ifm1¼1 orm2 ¼1.
Case 4G½VðGÞ ðSðGÞ [LðGÞÞffiC3¼ ðx1x2x3Þ.
If degðxiÞ 3 fori¼1;2;3, then the functionf :VðGÞ ! Pðf1;2gÞ defined by fðx1Þ ¼fðx2Þ ¼ ;;fðx3Þ ¼ f1g, and fðuÞ ¼ f2gotherwise, is a TR2DF of Tof weight at most n2, a contradiction. Henceforth, we may assume that degðx3Þ ¼2. Since Gis connected and dðGÞ ¼1, we may assume that degðx1Þ 3. Let G1;G2;. . .;Gk be the com- ponents ofG fx1;x2;x3g. Sincex1;x2;x3 are not support vertices, we have jVðGiÞj 2 for each i. If ctr2ðGiÞ jVðGiÞj 1 for somei, say i¼1, then letg be a ctr2ðG1Þ-function and define f :VðGÞ ! Pðf1;2gÞ by fðx1Þ ¼ ;;fðx2Þ ¼ f1g;fðx3Þ ¼ f2g;fðuÞ ¼gðuÞ for u2VðG1Þ, and fðuÞ ¼ f1g otherwise. Obviously, f is a TR2DF ofTof weight at mostn2, a contradiction. Thus, ctr2ðGiÞ ¼ jVðGiÞjfor eachi. As in Case 3, we can see that Gi2 Dfor eachi. IfjVðGiÞj 4 for somei, sayi¼1, and x1 is adjacent to a leaf ofG1 such as v, then the function f :VðGÞ ! Pðf1;2gÞ defined by fðx3Þ ¼fðvÞ ¼ ;;
fðx1Þ ¼ f1g, and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight at most n2, a contradiction. Thus, x1 is adjacent to a support vertex of some Gi. Similarly, x2 is adjacent to a support vertex of some Gj if degðx2Þ 3. It follows thatG2 D8.
Case 5G½VðGÞ ðSðGÞ [LðGÞÞffiC4¼ ðx1x2x3x4Þ.
Let G1;G2;. . .;Gk be the components of G fx1;x2;x3; x4g. Sincex1;x2;x3;x4are not support vertices,jVðGiÞj 2 for each i. Ifctr2ðGiÞ jVðGiÞj 1 for some i, sayi¼1, then let g be a ctr2ðG1Þ-function and define f :VðGÞ ! Pðf1;2gÞ by fðx1Þ ¼ ;;fðx2Þ ¼ f1g;fðx3Þ ¼fðx4Þ ¼ f2g;fðuÞ ¼gðuÞforu2VðG1Þ, andfðuÞ ¼ f1gotherwise.
Clearly,fis a TR2DF ofTof weight at mostn2 which is a contradiction. Thus,ctr2ðGiÞ ¼ jVðGiÞjfor each i. As in Case 3, we can see that Gi2 D for each i. If degðx1Þ 3;degðx2Þ 3, then the function f :VðGÞ ! Pðf1;2gÞ defined by fðx1Þ ¼fðx2Þ ¼ ;;fðx3Þ ¼ fðx4Þ ¼ f1g, and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight at most n2, a contradiction. If degðx1Þ 3;degðx3Þ 3, then the function f :VðGÞ ! Pðf1;2gÞ defined by fðx4Þ ¼fðx2Þ ¼ ;;fðx1Þ ¼ f1g;
fðx3Þ ¼ f2g, and fðuÞ ¼ f2g otherwise, is a TR2DF of T of weight at most n2, a contradiction. Henceforth, exactly one of thexis (i¼1;2;3) has degree greater than 2.
Assume that degðx1Þ 3. If jVðGiÞj 4 for some i, say i¼1, and x1 is adjacent to a leaf of G1 such as v, then the function f :VðGÞ ! Pðf1;2gÞ defined by
fðx4Þ ¼fðvÞ ¼ ;;fðx3Þ ¼ f1g, and fðuÞ ¼ f2g otherwise, is a TR2DF of Tof weight at mostn2, a contradiction.
Thus, x1 is adjacent to a support vertex of each Gi. This implies thatG2 D8. This completes the proof. h Considering Lemmas 6 and 11, we are now ready to state the main theorem of this paper.
Theorem 11 Let G be a connected graph of order n. Then, ctr2ðGÞ ¼n1 if and only if G2 fC3;C4;C5g [ ð[8i¼1DiÞ.
Acknowledgements The authors are grateful to anonymous referees for their remarks and suggestions that helped improve the manuscript.
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