Review Topical Sections

Shape memory alloy heat activation: State of the art review

  • Received: 31 August 2020 Accepted: 20 November 2020 Published: 03 December 2020
  • The use of shape memory alloys (SMAs) in civil structures attracted the attention of researchers worldwide. This interest is due the unique properties of SMAs, the shape memory effect (SME) and pseudoelasticity. Other desirable attributes for SMA include biocompatibility, high specific strength, wear resistance, anti-fatigue characteristics, and high yield strength. These beneficial qualities allow for a wide range of applications ranging from aerospace field to the medical and civil engineering fields. To intelligently utilize shape memory alloys and widen their application potential, certain characteristics including the heat activation and heat transfer mechanisms must be thoroughly understood. Such understanding would allow the development and optimization of systems utilizing SMA. Hence, this paper presents a state-of-the-art review about shape memory alloys with specific focus on the heat activation mechanisms and the heat transfer modes.

    Citation: Abdelmoneim El Naggar, Maged A. Youssef. Shape memory alloy heat activation: State of the art review[J]. AIMS Materials Science, 2020, 7(6): 836-858. doi: 10.3934/matersci.2020.6.836

    Related Papers:

    [1] Abel Cabrera Martínez, Iztok Peterin, Ismael G. Yero . Roman domination in direct product graphs and rooted product graphs. AIMS Mathematics, 2021, 6(10): 11084-11096. doi: 10.3934/math.2021643
    [2] Fu-Tao Hu, Xing Wei Wang, Ning Li . Characterization of trees with Roman bondage number 1. AIMS Mathematics, 2020, 5(6): 6183-6188. doi: 10.3934/math.2020397
    [3] Rangel Hernández-Ortiz, Luis Pedro Montejano, Juan Alberto Rodríguez-Velázquez . Weak Roman domination in rooted product graphs. AIMS Mathematics, 2021, 6(4): 3641-3653. doi: 10.3934/math.2021217
    [4] Mingyu Zhang, Junxia Zhang . On Roman balanced domination of graphs. AIMS Mathematics, 2024, 9(12): 36001-36011. doi: 10.3934/math.20241707
    [5] Jian Yang, Yuefen Chen, Zhiqiang Li . Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1. AIMS Mathematics, 2023, 8(8): 17702-17718. doi: 10.3934/math.2023904
    [6] Saeed Kosari, Yongsheng Rao, Zehui Shao, Jafar Amjadi, Rana Khoeilar . Complexity of signed total k-Roman domination problem in graphs. AIMS Mathematics, 2021, 6(1): 952-961. doi: 10.3934/math.2021057
    [7] Zhibin Du, Ayu Ameliatul Shahilah Ahmad Jamri, Roslan Hasni, Doost Ali Mojdeh . Maximal first Zagreb index of trees with given Roman domination number. AIMS Mathematics, 2022, 7(7): 11801-11812. doi: 10.3934/math.2022658
    [8] Bana Al Subaiei, Ahlam AlMulhim, Abolape Deborah Akwu . Vertex-edge perfect Roman domination number. AIMS Mathematics, 2023, 8(9): 21472-21483. doi: 10.3934/math.20231094
    [9] Chang-Xu Zhang, Fu-Tao Hu, Shu-Cheng Yang . On the (total) Roman domination in Latin square graphs. AIMS Mathematics, 2024, 9(1): 594-606. doi: 10.3934/math.2024031
    [10] Abel Cabrera-Martínez, Andrea Conchado Peiró . On the {2}-domination number of graphs. AIMS Mathematics, 2022, 7(6): 10731-10743. doi: 10.3934/math.2022599
  • The use of shape memory alloys (SMAs) in civil structures attracted the attention of researchers worldwide. This interest is due the unique properties of SMAs, the shape memory effect (SME) and pseudoelasticity. Other desirable attributes for SMA include biocompatibility, high specific strength, wear resistance, anti-fatigue characteristics, and high yield strength. These beneficial qualities allow for a wide range of applications ranging from aerospace field to the medical and civil engineering fields. To intelligently utilize shape memory alloys and widen their application potential, certain characteristics including the heat activation and heat transfer mechanisms must be thoroughly understood. Such understanding would allow the development and optimization of systems utilizing SMA. Hence, this paper presents a state-of-the-art review about shape memory alloys with specific focus on the heat activation mechanisms and the heat transfer modes.


    In this paper, we shall only consider graphs without multiple edges or loops. Let G=(V(G),E(G)) be a graph, vV(G), the neighborhood of v in G is denoted by N(v). That is to say N(v)={u|uvE(G),uV(G)}. The degree of a vertex v is denoted by d(v), i.e. d(v)=|N(v)|. A graph is trivial if it has a single vertex. The maximum degree and the minimum degree of a graph G are denoted by Δ(G) and δ(G), respectively. Denote by Kn the complete graph on n vertices.

    A subset D of the vertex set of a graph G is a dominating set if every vertex not in D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of G with |D|=γ(G) is called a γ-set of G.

    Roman domination of graphs is an interesting variety of domination, which was proposed by Cockayne et al. [6]. A Roman dominating function (RDF) of a graph G is a function f:V(G){0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight w(f) of a Roman dominating function f is the value w(f)=uV(G)f(u). The minimum weight of an RDF on a graph G is called the Roman domination number γR(G) of G. An RDF f of G with w(f)=γR(G) is called a γR-function of G. The problems on domination and Roman domination of graphs have been investigated widely, for example, see list of references [8,9,10,13] and [3,7,12], respectively.

    In 2016, Chellali et al. [5] introduced a variant of Roman dominating functions, called Roman {2}-dominating functions. A Roman {2}-dominating function (R{2}DF) of G is a function f:V{0,1,2} such that uN(v)f(u)2 for every vertex vV with f(v)=0. The weight of a Roman {2}-dominating function f is the sum vVf(v). The Roman {2}-domination number γ{R2}(G) is the minimum weight of an R{2}DF of G. Note that if f is an R{2}DF of G and v is a vertex with f(v)=0, then either there is a vertex uN(v) with f(u)=2, or at least two vertices x,yN(v) with f(x)=f(y)=1. Hence, an R{2}DF of G is also an RDF of G, which is also mentioned by Chellali et al [5]. Moreover, they showed that the decision problem for Roman {2}-domination is NP-complete, even for bipartite graphs.

    In fact, a Roman {2}-dominating function is essentially the same as a weak {2}-dominating function, which was introduced by Brešar et al. [1] and studied in literatures [2,11,14,15].

    For a mapping f:V(G){0,1,2}, let (V0,V1,V2) be the ordered partition of V(G) induced by f such that Vi={x:f(x)=i} for i=0,1,2. Note that there exists a 1-1 correspondence between the function f and the partition (V0,V1,V2) of V(G), so we will write f=(V0,V1,V2).

    Chellali et al. [4] obtained the following lower bound of Roman domination number.

    Lemma 1. (Chellali et al. [4]) Let G be a nontrivial connected graph with maximum degree Δ. Then γR(G)Δ+1Δγ(G).

    In this paper, we generalize this result on nontrivial connected graph G with maximum degree Δ and minimum degree δ. We prove that γR(G)Δ+2δΔ+δγ(G). As a corollary, we obtain that 32γ(G)γR(G)2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)2γ{R2}(G)1 for every graph G and there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k1 for any integer k2.

    Lemma 2. (Cockayne et al. [6]) Let f=(V0,V1,V2) be a γR-function of an isolate-free graph G with |V1| as small as possible. Then

    (i) No edge of G joins V1 and V2;

    (ii) V1 is independent, namely no edge of G joins two vertices in V1;

    (iii) Each vertex of V0 is adjacent to at most one vertex of V1.

    Theorem 3. Let G be a nontrivial connected graph with maximum degree Δ(G)=Δ and minimum degree δ(G)=δ. Then

    γR(G)Δ+2δΔ+δγ(G). (2.1)

    Moreover, if the equality holds, then

    γ(G)=n(Δ+δ)Δδ+Δ+δandγR(G)=n(Δ+2δ)Δδ+Δ+δ.

    Proof. Let f=(V0,V1,V2) be a γR-function of G with V1 as small as possible. By Lemma 2, we know that N(v)V0 for any vV1 and N(v1)N(v2)= for any v1,v2V1. So we have

    |V1||V0|δ (2.2)

    Since G is nontrivial, it follows that V2. Note that every vertex in V2 is adjacent to at most Δ vertices in V0; we have

    |V0|Δ|V2| (2.3)

    By Formulae (2.2) and (2.3), we have

    |V1|Δδ|V2| (2.4)

    By the definition of an RDF, every vertex in V0 has at least one neighbor in V2. So V1V2 is a dominating set of G. Together with Formula (2.4), we can obtain that

    γ(G)|V1|+|V2|Δδ|V2|+|V2|=Δ+δδ|V2|.

    Note that f is a γR-function of G; we have

    γR(G)=|V1|+2|V2|=(|V1|+|V2|)+|V2|γ(G)+δΔ+δγ(G)=Δ+2δΔ+δγ(G).

    Moreover, if the equality in Formula (2.1) holds, then by previous argument we obtain that |V1|=|V0|δ, |V0|=Δ|V2|, and V1V2 is a γ-set of G. Then we have

    n=|V0|+|V1|+|V2|=|V0|+|V0|δ+|V0|Δ=Δδ+Δ+δΔδ|V0|.

    Hence, we have

    |V0|=nΔδΔδ+Δ+δ,|V1|=nΔΔδ+Δ+δ, and |V2|=nδΔδ+Δ+δ.

    So

    γR(G)=|V1|+2|V2|=n(Δ+2δ)Δδ+Δ+δ and γ(G)=|V1|+|V2|=n(Δ+δ)Δδ+Δ+δ

    since V1V2 is a γ-set of G. This completes the proof.

    Now we show that the lower bound in Theorem 3 can be attained by constructing an infinite family of graphs. For any integers k2, δ2 and Δ=kδ, we construct a graph Hk from K1,Δ by adding k news vertices such that each new vertex is adjacent to δ vertices of K1,Δ with degree 1 and no two new vertices has common neighbors. Then add some edges between the neighbors of each new vertex u such that δ(Hk)=δ and the induced subetaaph of N(u) in Hk is not complete. The resulting graph Hk is a connected graph with maximum degree Δ(G)=Δ and maximum degree δ(G)=δ. It can be checked that γ(Hk)=k+1 and γR(Hk)=k+2=Δ+2δΔ+δγ(G).

    For example, if k=2, δ=3 and Δ=kδ=6, then the graph H2 constructed by the above method is shown in Figure 1, where u1 and u2 are new vertices.

    Figure 1.  An example to illustrate the construction of Hk.

    Furthermore, by Theorem 3, we can obtain a lower bound of the Roman domination number on regular graphs.

    Corollary 4. Let G be an r-regular graph, where r1. Then

    γR(G)32γ(G) (2.5)

    Moreover, if the equality holds, then

    γ(G)=2nr+2andγR(G)=3nr+2.

    Proof. Since G is r-regular, we have Δ(G)=δ(G)=r. By Theorem 3 we can obtain that this corollary is true.

    For any integer n2, denote by G2n the (2n2)-regular graph with 2n vertices, namely G2n is the graph obtained from K2n by deleting a perfect matching. It can be checked that γ(G2n)=2 and γR(G2n)=3=32γ(G) for any n2. Hence, the bound in Corollary 4 is attained.

    Note that γR(G)2γ(G) for any graph G; we can conclude the following result.

    Corollary 5. Let G be an r-regular graph, where r1. Then

    32γ(G)γR(G)2γ(G).

    Chellali et al. [5] obtain the following bounds for the Roman {2}-domination number of a graph G.

    Lemma 6. (Chellali et al. [5]) For every graph G, γ(G)γ{R2}(G)γR(G)2γ(G).

    Lemma 7. (Chellali et al. [5]) If G is a connected graph of order n and maximum degree Δ(G)=Δ, then

    γ{R2}(G)2nΔ+2.

    Theorem 8. For every graph G, γR(G)2γ{R2}(G)1. Moreover, for any integer k2, there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k1.

    Proof. Let f=(V0,V1,V2) be an γ{R2}-function of G. Then γ{R2}(G)=|V1|+2|V2| and γR(G)2|V1|+2|V2| since V1V2 is a dominating set of G. If |V2|1, then γR(G)2|V1|+2|V2|=2γ{R2}(G)2|V2|2γ{R2}(G)2. If |V2|=0, then every vertex in V0 is adjacent to at least two vertices in V1. So for any vertex uV1, f=(V0,{u},V1{u}) is an RDF of G. Then we have γR(G)1+2|V1{u}|=2|V1|1=2γ{R2}(G)1.

    For any integer k2, let Ik be the graph obtained from Kk by replacing every edge of Kk with two paths of length 2. Then Δ(Ik)=2(k1) and δ(Ik)=2. We first prove that γ{R2}(Ik)=k. Since V(Ik)=|V(Kk)|+2|E(Kk)|=k+2k(k1)2=k2, by Lemma 7 we can obtain γ{R2}(Ik)2|V(Ik)|Δ(Ik)+2=2k22(k1)+2=k. On the other hand, let f(x)=1 for each xV(Ik) with d(x)=2(k1) and f(y)=0 for each yV(Ik) with d(y)=2. It can be seen that f is an R{2}DF of Ik and w(f)=k. Hence, γ{R2}(Ik)=k.

    We now prove that γR(Ik)=2k1. Let g={V1,V2,V3} be a γR-function of Ik such that |V1| is minimum. For each 4-cycle C=v1v2v3v4v1 of Ik with d(v1)=d(v3)=2(k1) and d(v2)=d(v4)=2, we have wg(C)=g(v1)+g(v2)+g(v3)+g(v4)2. If wg(C)=2, then by Lemma 2(iii) we have g(vi){0,2} for any i{1,2,3,4}. Hence, one of v1 and v3 has value 2 and g(v2)=g(v4)=0. If wg(C)=3, then by Lemma 2(i) we have {g(v1),g(v3)}={1,2} or {g(v2),g(v4)}={1,2}. When {g(v2),g(v4)}={1,2}, let {g(v1),g(v2)}={1,2}, g(v2)=g(v4)=0 and g(x)=g(x) for any xV(Ik){v1,v2,v3,v4}. Then g is also a γR-function of Ik. If wg(C)=4, then exchange the values on C such that v1,v3 have value 2 and v2,v4 have value 0. So we obtain that Ik has a γR-function h such that h(y)=0 for any yV(Ik) with degree 2. Note that any two vertices of Ik with degree 2(k1) belongs to a 4-cycle considered above; we can obtain that there is exactly one vertex z of Ik with degree 2(k1) such that h(z)=1. Hence, γR(Ik)=w(h)=2k1.

    Note that the graph Ik constructed in Theorem 8 satisfies that γ(Ik)=k=γ{R2}(Ik). By Theorem 8, it suffices to prove that γ(Ik)=k. Let A={v:vV(Ik),d(v)=2(k1)} and B=V(Ik)A. We will prove that Ik has a γ-set containing no vertex of B. Let D be a γ-set of Ik. If D contains a vertex uB. Since the degree of u is 2, let u1 and u2 be two neighbors of u in Ik. Then d(u1)=d(u2)=2(k1) and, by the construction of Ik, u1 and u2 have two common neighbors u,u with degree 2. Hence, at least one of u,u1, and u2 belongs to D. Let D=(D{u,u}){u1,u2}. Then D is also a γ-set of Ik. Hence, we can obtain a γ-set of Ik containing no vertex of B by performing the above operation for each vertex vDB. So A is a γ-set of Ik and γ(Ik)=|A|=k.

    By Lemma 6 and Theorem 8, we can obtain the following corollary.

    Corollary 9. For every graph G, γ{R2}(G)γR(G)2γ{R2}(G)1.

    Theorem 10. For every graph G, γR(G)γ(G)+γ{R2}(G)1.

    Proof. By Lemma 6 we can obtain that γR(G)2γ(G)γ(G)+γ{R2}(G). If the equality holds, then γR(G)=2γ(G) and γ(G)=γ{R2}(G). So γR(G)=2γ{R2}(G), which contradicts Theorem 8. Hence, we have γR(G)γ(G)+γ{R2}(G)1.

    In this paper, we prove that γR(G)Δ+2δΔ+δγ(G) for any nontrivial connected graph G with maximum degree Δ and minimum degree δ, which improves a result obtained by Chellali et al. [4]. As a corollary, we obtain that 32γ(G)γR(G)2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)2γ{R2}(G)1 for every graph G and the bound is achieved. Although the bounds in Theorem 3 and Theorem 8 are achieved, characterizing the graphs that satisfy the equalities remain a challenge for further work.

    The author thanks anonymous referees sincerely for their helpful suggestions to improve this work. This work was supported by the National Natural Science Foundation of China (No.61802158) and Natural Science Foundation of Gansu Province (20JR10RA605).

    The author declares that they have no conflict of interest.



    [1] Lagoudas DC (2008) Shape Memory Alloys: Modeling and Engineering Applications, Springer Science & Business Media.
    [2] Bunget G (2011) BATMAV—A bio-inspired micro-aerial vehicle for flapping flight[PhD's Dissertation]. USA: North Carolina State University.
    [3] Duerig T, Stoeckel D, Johnson D (2002) SMA—smart materials for medical applications, European Workshop on Smart Structures in Engineering and Technology, Society of Photo-Optical Instrumentation Engineers, 7-15.
    [4] Kohl M, Popp M, Krevet B (2004) Shape memory micromechanisms for microvalve applications, Smart Structures and Materials 2004: Active Materials: Behavior and Mechanics, Society of Photo-Optical Instrumentation Engineers, 106-117.
    [5] Lan CC, Yang YN (2009) A computational design method for a shape memory alloy actuated compliant finger. J Mech Design 131: 021009.
    [6] Stepan LL, Levi DS, Carman GP (2005) A thin-film nitinol heart valve. J Biomech Eng-T ASME 127: 915-918.
    [7] Veeramani AS, Buckner GD, Owen SE, et al. (2008) Modeling the dynamic behavior of a shape memory alloy actuated catheter. Smart Mater Struct 17: 015037.
    [8] Epps JJ, Chopra I (2001) In-flight tracking of helicopter rotor blades using shape memory alloy actuators. Smart Mater Struct 10: 104-111.
    [9] Gravatt LM, Mabe JH, Calkins FT, et al. (2010) Characterization of varied geometry shape memory alloy beams, Industrial and Commercial Applications of Smart Structures Technologies 2010, Society of Photo-Optical Instrumentation Engineers, 7645U.
    [10] Costanza G, Tata ME (2020) Shape memory alloys for aerospace, recent developments, and new applications: A short review. Materials 13: 1856.
    [11] Abou‐Elfath H (2017) Evaluating the ductility characteristics of self‐centering buckling‐restrained shape memory alloy braces. Smart Mater Struct 26: 055020.
    [12] Carreras G, Casciati F, Casciati S, et al. (2011) Fatigue laboratory tests toward the design of SMA portico‐braces. Smart Struct Syst 7: 41-57.
    [13] McCormick J, DesRoches R, Fugazza D, et al. (2007) Seismic assessment of concentrically braced steel frames with shape memory alloy braces. J Struct Eng 133: 862-870.
    [14] Moradi S, Alam MS, Asgarian B (2014) Incremental dynamic analysis of steel frames equipped with NiTi shape memory alloy braces. Struct Design Tall Spec 23: 1406-1425.
    [15] Indirli M, Castellano M, Clemente P, et al. (2011) Demo-application of shape memory alloy devices: The rehabilitation of the S. Giorgio Church bell-tower, Smart Structures and Materials 2001: Smart Systems for Bridges, Structures, and Highways, Society of Photo-Optical Instrumentation Engineers, 262-272.
    [16] Castellano M, Indirli M, Martelli A (2001) UProgress of application, research and development and design guidelines for shape memory alloy devices for cultural heritage structures in Italy, Smart Structures and Materials, Society of Photo-Optical Instrumentation Engineers, 250-261.
    [17] Croci G (2001) Strengthening of the basilica of St francis of assissi after the September 1997 earthquake. Struct Eng Int 8: 56-58.
    [18] Andrawes B, Shin M, Wierschem N (2010) Active confinement of reinforced concrete bridge columns using shape memory alloys. J Bridge Eng 15: 81-89.
    [19] Hussain MA, Driver RG (2005) Experimental investigation of external confinement of reinforced concrete columns by hollow structural section collars. ACI Struct J 102: 242-251.
    [20] Krstulovic-Opara N, Thiedeman PD (2000) Active confinement of concrete members with self-stressing composites. ACI Mater J 97: 297-308.
    [21] Schmidt M, Ullrich J, Wieczorek A, et al. (2016) Experimental methods for investigation of shape memory based elastocaloric cooling processes and model validation. JoVE 111: e53626.
    [22] Vollach S, Shilo D (2010) The mechanical response of shape memory alloys under a rapid heating pulse. Exp Mech 50: 803-811.
    [23] Vollach S, Shilo D, Shlagman H (2016) Mechanical response of shape memory alloys under a rapid heating pulse-part Ⅱ. Exp Mech 56: 1465-1475.
    [24] Malka Y, Shilo D (2017) A fast and powerful release mechanism based on pulse heating of shape memory wires. Smart Mater Struct 26: 095061.
    [25] Motzki P, Gorges T, Kappel M, et al. (2018) High-speed and high-efficiency shape memory alloy actuation. Smart Mater Struct 27: 075047.
    [26] Barnes B, Brei D, Luntz J, Browne A, et al. (2006) Panel deployment using ultrafast SMA latches, ASME 2006 International Mechanical Engineering Congress and Exposition, 273-280.
    [27] Huang W (2002) On the selection of shape memory alloys for actuators. Mater Design 23: 11-19.
    [28] Leary M, Schiavone F, Subic A (2010) Lagging for control of shape memory alloy actuator response time. Mater Design 31: 2124-2128.
    [29] Van Humbeeck J (1999). Non-medical applications of shape memory alloys. Mater Sci Eng A-Struct 273: 134-148.
    [30] Leary M, Schiavone F, Subic A (2010) Lagging for control of shape memory alloy actuator response time. Mater Design 31: 2124-2128.
    [31] Huang S, Leary M, Ataalla T, et al. (2012) Optimisation of Ni-Ti shape memory alloy response time by transient heat transfer analysis. Mater Design 35: 655-663.
    [32] Morgan VT (1975) The overall convective heat transfer from smooth circular cylinders, In: Abraham JP, Gorman JM, Minkowycz WJ, Advances in Heat Transfer, Elsevier, 11: 199-264
    [33] Shahin AR, Meckl PH, Jones JD, et al. (1994) Enhanced cooling of shape memory alloy wires using semiconductor "heat pump" modules. J Intel Mat Syst Str 5: 95-104.
    [34] Pathak A, Brei D, Luntz J (2010) Transformation strain-based method for characterization of convective heat transfer from shape memory alloy wires. Smart Mater Struct 19: 035005.
    [35] Hilpert R (1933) Correlations for laminar forced convection in flow over an isothermal flat plate and in developing and fully developed flow in an isothermal tube. Forsch Ingenieurwes 4: 215-224
    [36] Eggeler G, Hornbogen E, Yawny A, et al. (2004) Structural and functional fatigue of NiTi shape memory alloys. Mater Sci Eng A-Struct 378: 24-33.
    [37] Costanza G, Paoloni S, Tata ME (2014) IR thermography and resistivity investigations on Ni-Ti shape memory alloy. Key Eng Mater 605: 23-26.
    [38] Dolce M, Cardone D (2001) Mechanical behaviour of shape memory alloys for seismic applications 1. Martensite and austenite NiTi bars subjected to torsion. Int J Mech Sci 43: 2631-2656.
    [39] Costanza G, Tata ME, Libertini R (2016) Effect of temperature on the mechanical behaviour of Ni-Ti shape memory sheets, TMS 2016 145th Annual Meeting & Exhibition, Springer, Cham, 433-439.
  • This article has been cited by:

    1. Chang-Xu Zhang, Fu-Tao Hu, Shu-Cheng Yang, On the (total) Roman domination in Latin square graphs, 2024, 9, 2473-6988, 594, 10.3934/math.2024031
    2. Sakander Hayat, Raman Sundareswaran, Marayanagaraj Shanmugapriya, Asad Khan, Venkatasubramanian Swaminathan, Mohamed Hussian Jabarullah, Mohammed J. F. Alenazi, Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling, 2024, 16, 2073-8994, 663, 10.3390/sym16060663
    3. Tatjana Zec, On the Roman domination problem of some Johnson graphs, 2023, 37, 0354-5180, 2067, 10.2298/FIL2307067Z
    4. Jian Yang, Yuefen Chen, Zhiqiang Li, Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1, 2023, 8, 2473-6988, 17702, 10.3934/math.2023904
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(7238) PDF downloads(848) Cited by(16)

Figures and Tables

Figures(15)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog