
Citation: Carol J. Farran, Olimpia Paun, Fawn Cothran, Caryn D. Etkin, Kumar B. Rajan, Amy Eisenstein, and Maryam Navaie. Impact of an Individualized Physical Activity Intervention on Improving Mental Health Outcomes in Family Caregivers of Persons with Dementia: A Randomized Controlled Trial[J]. AIMS Medical Science, 2016, 3(1): 15-31. doi: 10.3934/medsci.2016.1.15
[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 |
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[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 |
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In this paper, we shall only consider graphs without multiple edges or loops. Let $ G = (V(G), E(G)) $ be a graph, $ v \in V(G) $, the neighborhood of $ v $ in $ G $ is denoted by $ N(v) $. That is to say $ N(v) = \{u| uv \in E(G), u \in V(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 $ \Delta(G) $ and $ \delta(G) $, respectively. Denote by $ K_n $ 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 $ \gamma(G) $ is the minimum cardinality of a dominating set of $ G $. A dominating set $ D $ of $ G $ with $ |D| = \gamma(G) $ is called a $ \gamma $-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) \rightarrow \{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) = \sum_{u \in V(G)}f(u) $. The minimum weight of an RDF on a graph $ G $ is called the Roman domination number $ \gamma_R(G) $ of $ G $. An RDF $ f $ of $ G $ with $ w(f) = \gamma_{R}(G) $ is called a $ \gamma_{R} $-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\rightarrow \{0, 1, 2\} $ such that $ \sum_{u\in N(v)}f(u)\geq 2 $ for every vertex $ v\in V $ with $ f(v) = 0 $. The weight of a Roman {2}-dominating function $ f $ is the sum $ \sum_{v\in V}f(v) $. The Roman {2}-domination number $ \gamma_{\{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 $ u\in N(v) $ with $ f(u) = 2 $, or at least two vertices $ x, y \in N(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)\rightarrow \{0, 1, 2\} $, let $ (V_0, V_1, V_2) $ be the ordered partition of $ V(G) $ induced by $ f $ such that $ V_i = \{x: f(x) = i\} $ for $ i = 0, 1, 2 $. Note that there exists a 1-1 correspondence between the function $ f $ and the partition $ (V_0, V_1, V_2) $ of $ V(G) $, so we will write $ f = (V_0, V_1, V_2) $.
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 $ \Delta $. Then $ \gamma_{R}(G)\geq\frac{\Delta+1}{\Delta}\gamma(G) $.
In this paper, we generalize this result on nontrivial connected graph $ G $ with maximum degree $ \Delta $ and minimum degree $ \delta $. We prove that $ \gamma_{R}(G)\geq\frac{\Delta+2\delta}{\Delta+\delta}\gamma(G) $. As a corollary, we obtain that $ \frac{3}{2}\gamma(G)\leq\gamma_{R}(G)\leq 2\gamma(G) $ for any nontrivial regular graph $ G $. Moreover, we prove that $ \gamma_{R}(G)\leq2\gamma_{\{R2\}}(G)-1 $ for every graph $ G $ and there exists a graph $ I_k $ such that $ \gamma_{\{R2\}}(I_k) = k $ and $ \gamma_{R}(I_k) = 2k-1 $ for any integer $ k\geq2 $.
Lemma 2. (Cockayne et al. [6]) Let $ f = (V_0, V_1, V_2) $ be a $ \gamma_{R} $-function of an isolate-free graph $ G $ with $ |V_1| $ as small as possible. Then
(i) No edge of $ G $ joins $ V_1 $ and $ V_2 $;
(ii) $ V_1 $ is independent, namely no edge of $ G $ joins two vertices in $ V_1 $;
(iii) Each vertex of $ V_0 $ is adjacent to at most one vertex of $ V_1 $.
Theorem 3. Let $ G $ be a nontrivial connected graph with maximum degree $ \Delta(G) = \Delta $ and minimum degree $ \delta(G) = \delta $. Then
$ $
|
(2.1) |
Moreover, if the equality holds, then
$ \gamma(G) = \frac{n(\Delta+\delta)}{\Delta\delta+\Delta+\delta} \; \; \mathit{\text{and}}\; \; \gamma_R(G) = \frac{n(\Delta+2\delta)}{\Delta\delta+\Delta+\delta}. $ |
Proof. Let $ f = (V_0, V_1, V_2) $ be a $ \gamma_{R} $-function of $ G $ with $ V_1 $ as small as possible. By Lemma 2, we know that $ N(v)\subseteq V_0 $ for any $ v\in V_1 $ and $ N(v_1)\cap N(v_2) = \emptyset $ for any $ v_1, v_2\in V_1 $. So we have
$ $
|
(2.2) |
Since $ G $ is nontrivial, it follows that $ V_2\neq \emptyset $. Note that every vertex in $ V_2 $ is adjacent to at most $ \Delta $ vertices in $ V_0 $; we have
$ $
|
(2.3) |
By Formulae (2.2) and (2.3), we have
$ $
|
(2.4) |
By the definition of an RDF, every vertex in $ V_0 $ has at least one neighbor in $ V_2 $. So $ V_1\cup V_2 $ is a dominating set of $ G $. Together with Formula (2.4), we can obtain that
$ \gamma(G)\leq |V_1|+|V_2|\leq \frac{\Delta}{\delta}|V_2|+|V_2| = \frac{\Delta+\delta}{\delta}|V_2|. $ |
Note that $ f $ is a $ \gamma_{R} $-function of $ G $; we have
$ \gamma_R(G) = |V_1|+2|V_2| = (|V_1|+|V_2|)+|V_2|\geq \gamma(G)+\frac{\delta}{\Delta+\delta}\gamma(G) = \frac{\Delta+2\delta}{\Delta+\delta}\gamma(G). $ |
Moreover, if the equality in Formula (2.1) holds, then by previous argument we obtain that $ |V_1| = \frac{|V_0|}{\delta} $, $ |V_0| = \Delta|V_2| $, and $ V_1\cup V_2 $ is a $ \gamma $-set of $ G $. Then we have
$ n = |V_0|+|V_1|+|V_2| = |V_0|+\frac{|V_0|}{\delta}+\frac{|V_0|}{\Delta} = \frac{\Delta\delta+\Delta+\delta}{\Delta\delta}|V_0|. $ |
Hence, we have
$ |V_0| = \frac{n\Delta\delta}{\Delta\delta+\Delta+\delta}, \; \; |V_1| = \frac{n\Delta}{\Delta\delta+\Delta+\delta}, \;\text{ and }\; |V_2| = \frac{n\delta}{\Delta\delta+\Delta+\delta}. $ |
So
$ \gamma_R(G) = |V_1|+2|V_2| = \frac{n(\Delta+2\delta)}{\Delta\delta+\Delta+\delta} \;\text{ and }\; \gamma(G) = |V_1|+|V_2| = \frac{n(\Delta+\delta)}{\Delta\delta+\Delta+\delta} $ |
since $ V_1\cup V_2 $ is a $ \gamma $-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 $ k\geq 2 $, $ \delta\geq 2 $ and $ \Delta = k\delta $, we construct a graph $ H_k $ from $ K_{1, \Delta} $ by adding $ k $ news vertices such that each new vertex is adjacent to $ \delta $ vertices of $ K_{1, \Delta} $ 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 $ \delta(H_k) = \delta $ and the induced subetaaph of $ N(u) $ in $ H_k $ is not complete. The resulting graph $ H_k $ is a connected graph with maximum degree $ \Delta(G) = \Delta $ and maximum degree $ \delta(G) = \delta $. It can be checked that $ \gamma(H_k) = k+1 $ and $ \gamma_R(H_k) = k+2 = \frac{\Delta+2\delta}{\Delta+\delta}\gamma(G) $.
For example, if $ k = 2 $, $ \delta = 3 $ and $ \Delta = k\delta = 6 $, then the graph $ H_2 $ constructed by the above method is shown in Figure 1, where $ u_1 $ and $ u_2 $ are new vertices.
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 $ r\geq 1 $. Then
$ $
|
(2.5) |
Moreover, if the equality holds, then
$ \gamma(G) = \frac{2n}{r+2} \; \; \mathit{\text{and}}\; \; \gamma_R(G) = \frac{3n}{r+2}. $ |
Proof. Since $ G $ is $ r $-regular, we have $ \Delta(G) = \delta(G) = r $. By Theorem 3 we can obtain that this corollary is true.
For any integer $ n\geq 2 $, denote by $ G_{2n} $ the $ (2n-2) $-regular graph with $ 2n $ vertices, namely $ G_{2n} $ is the graph obtained from $ K_{2n} $ by deleting a perfect matching. It can be checked that $ \gamma(G_{2n}) = 2 $ and $ \gamma_R(G_{2n}) = 3 = \frac{3}{2}\gamma(G) $ for any $ n\geq 2 $. Hence, the bound in Corollary 4 is attained.
Note that $ \gamma_{R}(G)\leq2\gamma(G) $ for any graph $ G $; we can conclude the following result.
Corollary 5. Let $ G $ be an $ r $-regular graph, where $ r\geq 1 $. Then
$ $
|
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 $, $ \gamma(G)\leq\gamma_{\{R2\}}(G)\leq\gamma_{R}(G)\leq2\gamma(G) $.
Lemma 7. (Chellali et al. [5]) If $ G $ is a connected graph of order $ n $ and maximum degree $ \Delta(G) = \Delta $, then
$ \gamma_{\{R2\}}(G)\geq\frac{2n}{\Delta+2}. $ |
Theorem 8. For every graph $ G $, $ \gamma_{R}(G)\leq2\gamma_{\{R2\}}(G)-1 $. Moreover, for any integer $ k\geq2 $, there exists a graph $ I_k $ such that $ \gamma_{\{R2\}}(I_k) = k $ and $ \gamma_{R}(I_k) = 2k-1 $.
Proof. Let $ f = (V_0, V_1, V_2) $ be an $ \gamma_{\{R2\}} $-function of $ G $. Then $ \gamma_{\{R2\}}(G) = |V_1|+2|V_2| $ and $ \gamma_{R}(G)\leq 2|V_1|+2|V_2| $ since $ V_1\cup V_2 $ is a dominating set of $ G $. If $ |V_2|\geq 1 $, then $ \gamma_{R}(G)\leq 2|V_1|+2|V_2| = 2\gamma_{\{R2\}}(G)-2|V_2|\geq 2\gamma_{\{R2\}}(G)-2 $. If $ |V_2| = 0 $, then every vertex in $ V_0 $ is adjacent to at least two vertices in $ V_1 $. So for any vertex $ u\in V_1 $, $ f' = (V_0, \{u\}, V_1\setminus \{u\}) $ is an RDF of $ G $. Then we have $ \gamma_{R}(G)\leq 1+2|V_1\setminus \{u\}| = 2|V_1|-1 = 2\gamma_{\{R2\}}(G)-1 $.
For any integer $ k\geq2 $, let $ I_k $ be the graph obtained from $ K_k $ by replacing every edge of $ K_k $ with two paths of length 2. Then $ \Delta(I_k) = 2(k-1) $ and $ \delta(I_k) = 2 $. We first prove that $ \gamma_{\{R2\}}(I_k) = k $. Since $ V(I_k) = |V(K_k)|+2|E(K_k)| = k+2\cdot\frac{k(k-1)}{2} = k^2 $, by Lemma 7 we can obtain $ \gamma_{\{R2\}}(I_k)\geq\frac{2|V(I_k)|}{\Delta(I_k)+2} = \frac{2k^2}{2(k-1)+2} = k $. On the other hand, let $ f(x) = 1 $ for each $ x\in V(I_k) $ with $ d(x) = 2(k-1) $ and $ f(y) = 0 $ for each $ y\in V(I_k) $ with $ d(y) = 2 $. It can be seen that $ f $ is an R$ \{2\} $DF of $ I_k $ and $ w(f) = k $. Hence, $ \gamma_{\{R2\}}(I_k) = k $.
We now prove that $ \gamma_{R}(I_k) = 2k-1 $. Let $ g = \{V'_1, V'_2, V'_3\} $ be a $ \gamma_{R} $-function of $ I_k $ such that $ |V'_1| $ is minimum. For each 4-cycle $ C = v_1v_2v_3v_4v_1 $ of $ I_k $ with $ d(v_1) = d(v_3) = 2(k-1) $ and $ d(v_2) = d(v_4) = 2 $, we have $ w_g(C) = g(v_1)+g(v_2)+g(v_3)+g(v_4)\geq 2 $. If $ w_g(C) = 2 $, then by Lemma 2(iii) we have $ g(v_i)\in \{0, 2\} $ for any $ i\in\{1, 2, 3, 4\} $. Hence, one of $ v_1 $ and $ v_3 $ has value 2 and $ g(v_2) = g(v_4) = 0 $. If $ w_g(C) = 3 $, then by Lemma 2(i) we have $ \{g(v_1), g(v_3)\} = \{1, 2\} $ or $ \{g(v_2), g(v_4)\} = \{1, 2\} $. When $ \{g(v_2), g(v_4)\} = \{1, 2\} $, let $ \{g'(v_1), g'(v_2)\} = \{1, 2\} $, $ g'(v_2) = g'(v_4) = 0 $ and $ g'(x) = g(x) $ for any $ x\in V(I_k)\setminus \{v_1, v_2, v_3, v_4\} $. Then $ g' $ is also a $ \gamma_{R} $-function of $ I_k $. If $ w_g(C) = 4 $, then exchange the values on $ C $ such that $ v_1, v_3 $ have value 2 and $ v_2, v_4 $ have value 0. So we obtain that $ I_k $ has a $ \gamma_{R} $-function $ h $ such that $ h(y) = 0 $ for any $ y\in V(I_k) $ with degree 2. Note that any two vertices of $ I_k $ with degree $ 2(k-1) $ belongs to a 4-cycle considered above; we can obtain that there is exactly one vertex $ z $ of $ I_k $ with degree $ 2(k-1) $ such that $ h(z) = 1 $. Hence, $ \gamma_{R}(I_k) = w(h) = 2k-1 $.
Note that the graph $ I_k $ constructed in Theorem 8 satisfies that $ \gamma(I_k) = k = \gamma_{\{R2\}}(I_k) $. By Theorem 8, it suffices to prove that $ \gamma(I_k) = k $. Let $ A = \{v: v\in V(I_k), d(v) = 2(k-1)\} $ and $ B = V(I_k)\setminus A $. We will prove that $ I_k $ has a $ \gamma $-set containing no vertex of $ B $. Let $ D $ be a $ \gamma $-set of $ I_k $. If $ D $ contains a vertex $ u\in B $. Since the degree of $ u $ is 2, let $ u_1 $ and $ u_2 $ be two neighbors of $ u $ in $ I_k $. Then $ d(u_1) = d(u_2) = 2(k-1) $ and, by the construction of $ I_k $, $ u_1 $ and $ u_2 $ have two common neighbors $ u, u' $ with degree 2. Hence, at least one of $ u', u_1 $, and $ u_2 $ belongs to $ D $. Let $ D' = (D\setminus\{u, u'\})\cup\{u_1, u_2\} $. Then $ D' $ is also a $ \gamma $-set of $ I_k $. Hence, we can obtain a $ \gamma $-set of $ I_k $ containing no vertex of $ B $ by performing the above operation for each vertex $ v\in D\cap B $. So $ A $ is a $ \gamma $-set of $ I_k $ and $ \gamma(I_k) = |A| = k $.
By Lemma 6 and Theorem 8, we can obtain the following corollary.
Corollary 9. For every graph $ G $, $ \gamma_{\{R2\}}(G)\leq\gamma_{R}(G)\leq2\gamma_{\{R2\}}(G)-1 $.
Theorem 10. For every graph $ G $, $ \gamma_{R}(G)\leq\gamma(G)+\gamma_{\{R2\}}(G)-1 $.
Proof. By Lemma 6 we can obtain that $ \gamma_{R}(G)\leq2\gamma(G)\leq \gamma(G)+\gamma_{\{R2\}}(G) $. If the equality holds, then $ \gamma_{R}(G) = 2\gamma(G) $ and $ \gamma(G) = \gamma_{\{R2\}}(G) $. So $ \gamma_{R}(G) = 2\gamma_{\{R2\}}(G) $, which contradicts Theorem 8. Hence, we have $ \gamma_{R}(G)\leq\gamma(G)+\gamma_{\{R2\}}(G)-1 $.
In this paper, we prove that $ \gamma_{R}(G)\geq\frac{\Delta+2\delta}{\Delta+\delta}\gamma(G) $ for any nontrivial connected graph $ G $ with maximum degree $ \Delta $ and minimum degree $ \delta $, which improves a result obtained by Chellali et al. [4]. As a corollary, we obtain that $ \frac{3}{2}\gamma(G)\leq\gamma_{R}(G)\leq 2\gamma(G) $ for any nontrivial regular graph $ G $. Moreover, we prove that $ \gamma_{R}(G)\leq2\gamma_{\{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] |
Alzheimer's Association (2015) 2015 Alzheimer's disease facts and figures. Alzheimer’s Dement 11: 332-384. doi: 10.1016/j.jalz.2015.02.003
![]() |
[2] |
Gitlin L, Marx K, Stanley I, et al. (2015) Translating evidence-based dementia caregiving interventions into practice: State-of-the-science and next steps. Gerontologist 55: 210-226. doi: 10.1093/geront/gnu123
![]() |
[3] | Vernooij-Dassen J, Draskovic I, McCleery J, et al. (2011) Cognitive reframing for carers of people with dementia (Review). The Cochrane Collaboration, The Cochrane Library, 11, Available from: http://www.thecochranelibrary.com. |
[4] | Sörensen S, Pinquart M, Duberstein P (2002) How effective are interventions with caregivers? An updated meta-analysis. Gerontologist 42: 356-372. |
[5] |
Schulz R, O’Brien A, Czaja S, et al. (2002) Dementia caregiver intervention research: In search of clinical significance. Gerontologist 42: 589-602. doi: 10.1093/geront/42.5.589
![]() |
[6] | Kim H, Chang M, Rose K, et al. (2012) Predictors of caregiver burden in caregivers of individuals with dementia.J Adv Nurs68: 846-855. |
[7] | Adelman R, Tmanova L, Delgado D, et al. (2014) Caregiver burden: A clinical review.JAMA 311: 1052-1059. |
[8] |
Orgeta V, Sterzo E (2013) Sense of coherence, burden, and affective symptoms in family carers of people with dementia.Int Psychogeriatr 25: 973-980. doi: 10.1017/S1041610213000203
![]() |
[9] | Varela G, Varona L, Anderson K, et al. (2011) Alzheimer's care at home: A focus on caregivers strain. [L'impegno dei caregivers nell'assistenza a casa ai pazienti con malati di Alzheimer.] Prof Inferm 64: 113-117. |
[10] | Han J, Jeong H, Park J, et al. (2014) Effects of social supports on burden in caregivers of people with dementia.Int Psychogeriatr26: 1639-1648. |
[11] | Rodakowski J, Skidmore E, Rogers J, et al. (2012) Role of social support in predicting caregiver burden.Arch Phys Med Rehabil93: 2229-2236. |
[12] | Van der Lee J, Bakker T, Duivenvoorden H, et al. (2014) Multivariate models of subjective caregiver burden in dementia: A systematic review.Ageing Res Rev15: 76-93. |
[13] |
Ornstein K, Gaugler J (2012) The problem with "problem behaviors": A systematic review of the association between individual patient behavioral and psychological symptoms and caregiver depression and burden within the dementia patient-caregiver dyad.Int Psychogeriatr 24: 1536-1552. doi: 10.1017/S1041610212000737
![]() |
[14] | Givens J, Mezzacappa C, Heeren T, et al. (2014) Depressive symptoms among dementia caregivers: Role of mediating factors.Am J Geriatr Psychiatry22: 481-488. |
[15] | Schulz R, Sherwood P (2008) Physical and mental health effects of family caregiving. Am J Nurs 108: 23-27. |
[16] | Sörensen S, Pinquart M, Duberstein P (2002) How effective are interventions with caregivers? Gerontologist 42: 356-372. |
[17] | Livingston G, Barber J, Rapaport P, et al. (2014) START (STrAtegies for RelaTives) study: A pragmatic randomized controlled trial to determine the clinical effectiveness and cost-effectiveness of a manual-based coping strategy programme in promoting the mental health of careers of people with dementia. Hlth Tech Asst 18: 61. |
[18] | Thompson C, Spilsbury K, Hill J, et al. (2007) Systematic review of information and support interventions for caregivers of people with dementia. BMC Geriatr 7:1-12. |
[19] | de Labra C, Millán J, Buján A, et al. (2015) Predictors of caregiving satisfaction in informal caregivers of people with dementia. Arch Geront Geriat 60: 380-388. |
[20] | Mittelman M, Haley W, Clay O, et al. (2006) Improving caregiver well-being delays nursing home placement of patients with Alzheimer disease. Neurol 67: 1592-1599. |
[21] | Roth D, Dilworth-Anderson P, Huang J, et al. (2015) Positive aspects of family caregiving for dementia: Differential item functioning by race. J Gerontol: PSS 1-7. |
[22] | Fredman L, Gordon S, Heeren T, et al. (2013) Positive affect is associated with fewer sleep problems in older caregivers but not noncaregivers. Gerontologist 54: 559-569. |
[23] |
Lawton M, Kleban M, Moss M, et al. (1989) Measuring caregiving appraisal. J Gerontol 44: P61-71. doi: 10.1093/geronj/44.3.P61
![]() |
[24] | Bandura A (2002) Self-efficacy: The exercise of control. New York: WH Freeman. |
[25] | Roth D, Fredman L, Haley W (2015) Informal caregiving and its impact on health: A reappraisal from population-based studies. Gerontologist 55: 309-319. |
[26] | Zarit S, Kim K, Femia E, et al. (2013) The effects of adult day services on family caregivers’ daily stress, affect, and health: Outcomes from the daily stress and health (DaSH) Study. Gerontologist 54: 570-579. |
[27] | Physical Activity Guidelines Advisory Committee (2008) Physical Activity Guidelines Advisory Committee Report, 2008. Washington, DC: U.S. Department of Health and Human Services. |
[28] | Pinquart M, Sörensen S (2007) Correlates of physical health of informal caregivers: A meta-analysis. Gerontol B Psychol Sci Soc Sci 62: P126-137. |
[29] | Connell C, Janevic M (2009) Effects of a telephone-based exercise intervention for dementia caregiving wives: A randomized controlled trial. J App Gerontol 28: 171-194. |
[30] | King A, Baumann K, O'Sullivan P, et al. (2002) Effects of moderate-intensity exercise on physiological, behavioral, and emotional responses to family caregiving: A randomized controlled trial. J Gerontol A Biol Sci Med Sci 57: M26-36. |
[31] |
Castro C, Wilcox S, O'Sullivan P, et al. (2002) An exercise program for women who are caring for relatives with dementia. Psychosom Med 64: 458-468. doi: 10.1097/00006842-200205000-00010
![]() |
[32] | Orgeta V, Miranda-Castillo C (2014) Does physical activity reduce burden in carers of people with dementia? A literature review. Int J Geriatr Psychiatry 29: 771-783. |
[33] |
McAuley E, Elavsky S, Jerome G, et al. (2005) Physical activity-related well-being in older adults: Social cognitive influences. Psychol Aging 20: 295-302. doi: 10.1037/0882-7974.20.2.295
![]() |
[34] |
Elavsky S, McAuley E, Motl R, et al. (2005) Physical activity enhances long-term quality of life in older adults: Efficacy, esteem, and affective influences. Ann Behav Med 30:138-145. doi: 10.1207/s15324796abm3002_6
![]() |
[35] |
Farran C, Staffileno B, Gilley D, et al. (2008) A lifestyle physical activity intervention for caregivers of persons with Alzheimer's disease. Am J Alzheimers Dis Other Demen 23: 132-142. doi: 10.1177/1533317507312556
![]() |
[36] | National Institutes of Health, Office of Behavioral and Social Sciences Research, Randomized Clinical Trials Involving Behavioral Interventions (2015, November 15) https://obssr.od.nih.gov/training_and_education/annual_Randomzied_Clinical_Trials_course/RCT_info.aspx |
[37] |
Schulz R, Beach SR (1999) Caregiving as a risk factor for mortality: The caregiver health effects study. JAMA 282: 2215-2219 doi: 10.1001/jama.282.23.2215
![]() |
[38] | Farran C, Gilley D, McCann J, et al. (2004) Psychosocial interventions to reduce depressive symptoms of dementia caregivers: A randomized clinical trial comparing two approaches. J Ment Health Aging 10: 337-350. |
[39] | Cornoni-Huntley J, Brock D, Ostfeld A, et al. (1986) Established populations for epidemiologic studies of the elderly resource data book. (Rep. No. NIH Pub No. 86-2443). Washington, DC: U.S. Department of Health and Human Services. |
[40] | Folstein M, Folstein S, McHugh P (1975) Mini-mental state: A practical method for grading the mental state of patients for the clinician. J Psychiatr Res 12: 188-189. |
[41] | Stewart A, Mills K, King A, et al. (2001) CHAMPS physical activity questionnaire for older adults: Outcomes for interventions. Med Sci Sports and Exerc 33: 1126-1141. |
[42] |
Kohout F, Berkman L, Evans D, et al. (1993) Two shorter forms of the CES-D (Center for Epidemiological Studies Depression) depression symptoms index. J Aging Health 5: 179-193. doi: 10.1177/089826439300500202
![]() |
[43] |
Watson D, Clark L, Tellegen A (1988) Development and validation of brief measures of positive and negative affect: The PANAS scales. J Pers Soc Psychol 54: 1063-1070. doi: 10.1037/0022-3514.54.6.1063
![]() |
[44] | SAS 9.3, SAS Institute, Cary, North Carolina. |
[45] |
Orgeta V, Miranda-Castillo C (2014) Does physical activity reduce burden in carers of people with dementia? A literature review. Int J Geriatr Psychiatry 29: 771-783. doi: 10.1002/gps.4060
![]() |
[46] | Stromeyer E, Ward R (2012) Target populations, recruitment, retention, and optimal testing methods: Methodological issues for studies in the epidemiology of aging, In: Newman AB, Cauley JA (Eds), The Epidemiology of Aging, Springer, 49-68. |
[47] |
Samelson E, Kelsey J, Kiel D, et al. (2008) Issues in conducting epidemiologic research among elders: Lessons from the MOBILIZE Boston study. Am J Epidem 168: 1444-1451. doi: 10.1093/aje/kwn277
![]() |
[48] | Hill K, Smith R, Fearn M, et al. (2007) Physical and psychological outcomes of a supported physical activity program for older carers. J Aging Phys Act 15: 257-271. |
[49] |
Hirano A, Suzuki Y, Kuzuya M, et al. (2011) Influence of regular exercise on subjective sense of burden and physical symptoms in community-dwelling caregivers of dementia patients: A randomized controlled trial. Arch Gerontol Geriatr 53: e158-163. doi: 10.1016/j.archger.2010.08.004
![]() |
[50] |
Dahlrup B, Nordell E, Carlsson KS, et al. (2014) Health economic analysis on a psychosocial intervention for family caregivers of persons with dementia. Dement Geriatr Cogn Disord 37: 181-195. doi: 10.1159/000355365
![]() |
[51] |
Forster A, Dickerson J, Young J, et al. (2013) A structured training programme for caregivers of inpatients after stroke (TRACS): A cluster randomised controlled trial and cost-effectiveness analysis. The Lancet 382: 2069-2076. doi: 10.1016/S0140-6736(13)61603-7
![]() |
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