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Artykuły w czasopismach na temat „Signed domination”

Autor: Grafiati

Data publikacji: 3 czerwca 2025

Data aktualizacji: 31 lipca 2025

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1

Joseph, James, and Mayamma Joseph. "On domination in signed graphs." Acta Universitatis Sapientiae, Informatica 15, no. 1 (2023): 1–9. http://dx.doi.org/10.2478/ausi-2023-0001.

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Abstract In this article the concept of domination in signed graphs is examined from an alternate perspective and a new definition of the same is introduced. A vertex subset D of a signed graph S is a dominating set, if for each vertex v not in D there exists a vertex u ∈ D such that the sign of the edge uv is positive. The domination number γ (S) of S is the minimum cardinality among all the dominating sets of S. We obtain certain bounds of γ (S) and present a necessary and su cient condition for a dominating set to be a minimal dominating set. Further, we characterise the signed graphs havin

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2

Mathias, Anisha Jean, V. Sangeetha, and Mukti Acharya. "Restrained domination in signed graphs." Acta Universitatis Sapientiae, Mathematica 12, no. 1 (2020): 155–63. http://dx.doi.org/10.2478/ausm-2020-0010.

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AbstractA signed graph Σ is a graph with positive or negative signs attatched to each of its edges. A signed graph Σ is balanced if each of its cycles has an even number of negative edges. Restrained dominating set D in Σ is a restrained dominating set of its underlying graph where the subgraph induced by the edges across Σ[D : V \ D] and within V \ D is balanced. The set D having least cardinality is called minimum restrained dominating set and its cardinality is the restrained domination number of Σ denoted by γr(Σ). The ability to communicate rapidly within the network is an important appli

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3

Patil, Dr. Mallikarjun S. "A study on signed domination in graphs in present context – A theoretical assimilation." International Journal of Advance and Applied Research 4, no. 33 (2023): 97–104. https://doi.org/10.5281/zenodo.10153459.

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<strong>Abstract:</strong>The study of signed domination in graphs has gained significant attention in recent years, owing to its relevance in various real-world applications, including social networks, communication systems, and optimization problems. In this research article, we provide a contemporary overview of signed domination in graphs, focusing on its current context and emerging trends. Signed domination, a generalization of classical domination in graph theory, involves assigning positive or negative weights to vertices in a graph. A set of vertices is said to be a signed dominating

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4

Almulhim, Ahlam. "Signed double Italian domination." AIMS Mathematics 8, no. 12 (2023): 30895–909. http://dx.doi.org/10.3934/math.20231580.

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<abstract><p>A signed double Italian dominating function (SDIDF) on a graph $ G = (V, E) $ is a function $ f $ from $ V $ to $ \{-1, 1, 2, 3\} $, satisfying (ⅰ) $ \sum_{u\in N[v]}f(u)\ge1 $ for all $ v\in V $; (ⅱ) if $ f(v) = -1 $ for some $ v\in V $, then there exists $ A\subseteq N(v) $ such that $ \sum_{u\in A}f(u)\ge3 $; and (ⅲ) if $ f(v) = 1 $ for some $ v\in V $, then there exists $ A\subseteq N(v) $ such that $ \sum_{u\in A}f(u)\ge2 $. The weight of an SDIDF $ f $ is $ \sum_{v\in V}f(v) $. The signed double Italian domination number of $ G $ is the minimum weight of an SDIDF

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5

Ghameshlou, Arezoo N., Athena Shaminezhad, Ebrahim Vatandoost, and Abdollah Khodkar. "Signed domination and Mycielski’s structure in graphs." RAIRO - Operations Research 54, no. 4 (2020): 1077–86. http://dx.doi.org/10.1051/ro/2019109.

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Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete

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6

Dehgardi, Nasrin, Maryam Atapour, and Abdollah Khodkar. "Twin signed k-domination numbers in directed graphs." Filomat 31, no. 20 (2017): 6367–78. http://dx.doi.org/10.2298/fil1720367d.

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Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers

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7

Xu, Baogen, Mengmeng Zheng, and Ting Lan. "On Non-Zero Vertex Signed Domination." Symmetry 15, no. 3 (2023): 741. http://dx.doi.org/10.3390/sym15030741.

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For a graph G=(V,E) and a function f:V→{−1,+1}, if S⊆V then we write f(S)=∑v∈Sf(v). A function f is said to be a non-zero vertex signed dominating function (for short, NVSDF) of G if f(N[v])=0 holds for every vertex v in G, and the non-zero vertex signed domination number of G is defined as γsb(G)=max{f(V)|f is an NVSDF of G}. In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of γsb(G) for s

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8

Sheikholeslami, Seyed Mahmoud, Rana Khoeilar, and Leila Asgharsharghi. "Signed strong Roman domination in graphs." Tamkang Journal of Mathematics 48, no. 2 (2017): 135–47. http://dx.doi.org/10.5556/j.tkjm.48.2017.2240.

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Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong Roman dominating function (abbreviated SStRDF) on a graph $G$ is a function $f:V\to \{-1,1,2,\ldots,\lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the conditions that (i) for every vertex $v$ of $G$, $\sum_{u\in N[v]} f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $u$ for which $f(u)\ge 1+\lceil\frac{1}{2}|N(u)\cap V_{-1}|\rceil$, where $V_{-1}=\{v\in V \mid f(v)=-1\}$. The minimum of the values $\sum_{v\

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9

Xu, Baogen, Ting Lan, and Mengmeng Zheng. "A Note on the Signed Clique Domination Numbers of Graphs." Journal of Mathematics 2022 (September 17, 2022): 1–6. http://dx.doi.org/10.1155/2022/3208164.

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Let G = V , E be a graph. A function f : E ⟶ − 1 , + 1 is said to be a signed clique dominating function (SCDF) of G if ∑ e ∈ E K f e ≥ 1 holds for every nontrivial clique K in G . The signed clique domination number of G is defined as γ scl ′ G = min ∑ e ∈ E G f e | f is an SCDF of G . In this paper, we investigate the signed clique domination numbers of join of graphs. We correct two wrong results reported by Ao et al. (2014) and Ao et al. (2015) and determine the exact values of the signed clique domination numbers of P m ∨ K n ¯ and C m ∨ K n .

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10

Wang, Haichao, and Hye Kyung Kim. "Signed Domination Number of the Directed Cylinder." Symmetry 11, no. 12 (2019): 1443. http://dx.doi.org/10.3390/sym11121443.

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In a digraph D = ( V ( D ) , A ( D ) ) , a two-valued function f : V ( D ) → { - 1 , 1 } defined on the vertices of D is called a signed dominating function if f ( N - [ v ] ) ≥ 1 for every v in D. The weight of a signed dominating function is f ( V ( D ) ) = ∑ v ∈ V ( D ) f ( v ) . The signed domination number γ s ( D ) is the minimum weight among all signed dominating functions of D. Let P m × C n be the Cartesian product of directed path P m and directed cycle C n . In this paper, the exact value of γ s ( P m × C n ) is determined for any positive integers m and n.

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11

Mahmoodi, A. "On the signed strong Roman domination number of graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (2020): 2050028. http://dx.doi.org/10.1142/s1793830920500287.

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Let [Formula: see text] be a finite and simple graph of order [Formula: see text] and maximum degree [Formula: see text]. A signed strong Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) for every vertex [Formula: see text] of [Formula: see text], [Formula: see text], where [Formula: see text] is the closed neighborhood of [Formula: see text] and (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text], where [Formula: see

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12

Tang, Huajun, and Yaojun Chen. "Upper signed domination number." Discrete Mathematics 308, no. 15 (2008): 3416–19. http://dx.doi.org/10.1016/j.disc.2007.06.031.

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13

Zelinka, Bohdan. "Signed 2-domination in caterpillars." Mathematica Bohemica 129, no. 4 (2004): 393–98. http://dx.doi.org/10.21136/mb.2004.134049.

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14

CHELVAM, T. TAMIZH, G. KALAIMURUGAN, and WELL Y. CHOU. "THE SIGNED STAR DOMINATION NUMBER OF CAYLEY GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 02 (2012): 1250017. http://dx.doi.org/10.1142/s1793830912500176.

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Let G be a simple connected graph with vertex set V(G) and edge set E(G). A function f : E(G) → {-1, 1} is called a signed star dominating function (SSDF) on G if ∑e∈E(v) f(e) ≥ 1 for every v ∈ V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as γ SS (G) = min {∑e∈E(G) f(e) | f is a SSDF on G}. In this paper, we obtain exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs.

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15

Volkmann, Lutz. "Signed domination and signed domatic numbers of digraphs." Discussiones Mathematicae Graph Theory 31, no. 3 (2011): 415. http://dx.doi.org/10.7151/dmgt.1555.

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16

Germina, K. A., and P. K. Ashraf. "On open domination and domination in signed graphs." International Mathematical Forum 8 (2013): 1863–72. http://dx.doi.org/10.12988/imf.2013.310198.

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17

Shahbazi, L., H. Abdollahzadeh Ahangar, R. Khoeilar, and S. M. Sheikholeslami. "Signed total double Roman k-domination in graphs." Discrete Mathematics, Algorithms and Applications 12, no. 01 (2019): 2050009. http://dx.doi.org/10.1142/s1793830920500093.

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A signed total double Roman [Formula: see text]-dominating function (STDRkDF) on an isolated-free graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] has at least two neighbors assigned 2 under [Formula: see text] or at least one neighbor [Formula: see text] with [Formula: see text], (ii) every vertex [Formula: see text] with [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text] and (iii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STDRkDF is t

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18

Zhao, Yancai, and Zuosong Liang. "SIGNED STAR DOMINATION IN GRAPHS." Advances and Applications in Discrete Mathematics 26, no. 1 (2021): 17–33. http://dx.doi.org/10.17654/dm026010017.

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19

Li, Wen Sheng. "Negative Signed Domination in Digraphs." Applied Mechanics and Materials 65 (June 2011): 145–47. http://dx.doi.org/10.4028/www.scientific.net/amm.65.145.

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The Concept of Negative Signed Domination Number of a Directed Graph Is Introduced. Exact Values Are Found for the Directed Cycle and Particular Types of Tournaments. Furthermore, it Is Proved that the Negative Signed Domination Number May Be Arbitrarily Big for Digraphs with a Directed Hamiltonian Cycle.

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20

Ashraf, P. K., K. A. Germina, and Lishan Liu. "Double domination in signed graphs." Cogent Mathematics 3, no. 1 (2016): 1186135. http://dx.doi.org/10.1080/23311835.2016.1186135.

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21

Favaron, Odile. "Signed domination in regular graphs." Discrete Mathematics 158, no. 1-3 (1996): 287–93. http://dx.doi.org/10.1016/0012-365x(96)00026-x.

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22

Henning, Michael A. "Signed total domination in graphs." Discrete Mathematics 278, no. 1-3 (2004): 109–25. http://dx.doi.org/10.1016/j.disc.2003.06.002.

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23

Abdollahzadeh Ahangar, H., Michael A. Henning, Christian Löwenstein, Yancai Zhao, and Vladimir Samodivkin. "Signed Roman domination in graphs." Journal of Combinatorial Optimization 27, no. 2 (2012): 241–55. http://dx.doi.org/10.1007/s10878-012-9500-0.

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Sheikholeslami, S. M., and L. Volkmann. "Signed Roman domination in digraphs." Journal of Combinatorial Optimization 30, no. 3 (2013): 456–67. http://dx.doi.org/10.1007/s10878-013-9648-2.

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Sheikholeslami, Seyed Mahmoud, Asghar Bodaghli, and Lutz Volkmann. "Twin signed Roman domination numbers in directed graphs." Tamkang Journal of Mathematics 47, no. 3 (2016): 357–71. http://dx.doi.org/10.5556/j.tkjm.47.2016.2035.

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Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin

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26

Amjadi, J., and M. Soroudi. "Twin signed total Roman domination numbers in digraphs." Asian-European Journal of Mathematics 11, no. 03 (2018): 1850034. http://dx.doi.org/10.1142/s1793557118500341.

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Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text] and arc set [Formula: see text]. A twin signed total Roman dominating function (TSTRDF) on the digraph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] and [Formula: see text] for each [Formula: see text], where [Formula: see text] (respectively [Formula: see text]) consists of all in-neighbors (respectively out-neighbors) of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an in-neighbor [Formula: see

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27

LEE, CHUAN-MIN, and CHENG-CHIEN LO. "On the Complexity of Reverse Minus and Signed Domination on Graphs." Journal of Interconnection Networks 15, no. 01n02 (2015): 1550008. http://dx.doi.org/10.1142/s0219265915500085.

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Motivated by the concept of reverse signed domination, we introduce the reverse minus domination problem on graphs, and study the reverse minus and signed domination problems from the algorithmic point of view. In this paper, we show that both the reverse minus and signed domination problems are polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs, and are linear-time solvable for trees. For chordal graphs and bipartite planar graphs, however, we show that the decision problem corresponding to the reverse minus domination problem is NP-complete. For doubly chorda

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28

PRADHAN, D. "COMPLEXITY OF CERTAIN FUNCTIONAL VARIANTS OF TOTAL DOMINATION IN CHORDAL BIPARTITE GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 03 (2012): 1250045. http://dx.doi.org/10.1142/s1793830912500450.

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In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD ⊆ V of a given graph G = (V, E) such that |TD∩NG(v)| ≥ 1 for every v ∈ V. In the minimum signed (minus) total domination problem for a graph G = (V, E), it is required to find a function f : V → {-1, 1} ({-1, 0, 1}) such that f(NG(v)) = ∑u∈NG(v)f(u) ≥ 1 for each v ∈ V, and the cost f(V) = ∑v∈V

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Zelinka, Bohdan. "On signed edge domination numbers of trees." Mathematica Bohemica 127, no. 1 (2002): 49–55. http://dx.doi.org/10.21136/mb.2002.133984.

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Hong, Xia, Tianhu Yu, Zhengbang Zha, and Huihui Zhang. "The signed Roman domination number of two classes graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (2020): 2050024. http://dx.doi.org/10.1142/s179383092050024x.

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Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A signed Roman dominating function (SRDF) of [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] for each [Formula: see text], where [Formula: see text] is the set, called closed neighborhood of [Formula: see text], consists of [Formula: see text] and the vertex of [Formula: see text] adjacent to [Formula: see text] (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula:

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Yang, Hong, Pu Wu, Sakineh Nazari-Moghaddam, et al. "Bounds for signed double Roman k-domination in trees." RAIRO - Operations Research 53, no. 2 (2019): 627–43. http://dx.doi.org/10.1051/ro/2018043.

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Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this

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Sankar, Chakaravarthy, Chandran Kalaivani, Perumal Chellamani, and Gangatharan Venkat Narayanan. "Analyzing Network Stability via Symmetric Structures and Domination Integrity in Signed Fuzzy Graphs." Symmetry 17, no. 5 (2025): 766. https://doi.org/10.3390/sym17050766.

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The concept of domination is introduced within the context of signed fuzzy graphs (signed-FGs), along with examples, as a novel metric to evaluate graph stability under varying conditions. This metric particularly focuses on dominant sets and integrity measures, providing a well-rounded approach to assessing the structural stability of signed- FGs. The necessity of fulfilling the domination integrity condition in evaluating the performance of signed-FGs is highlighted through a discussion on its formulation and an analysis of its upper and lower bounds. An algorithm for identifying strong arcs

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ATAPOUR, MARYAM, SEYED MAHMOUD SHEIKHOLESLAMI, and ABDOLLAH KHODKAR. "Global signed total domination in graphs." Publicationes Mathematicae Debrecen 79, no. 1-2 (2011): 7–22. http://dx.doi.org/10.5486/pmd.2011.4826.

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R, Malathi, and Alwin A. "Domination Number on Balanced Signed Graphs." INTERNATIONAL JOURNAL OF COMPUTING ALGORITHM 3, no. 3 (2014): 324–27. http://dx.doi.org/10.20894/ijcoa.101.003.003.038.

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Volkmann, Lutz. "Signed total Roman domination in digraphs." Discussiones Mathematicae Graph Theory 37, no. 1 (2017): 261. http://dx.doi.org/10.7151/dmgt.1929.

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Alikhani, Saeid, and Fatemeh Ramezani. "Signed Domination Number of Some Graphs." Iranian Journal of Science and Technology, Transactions A: Science 46, no. 1 (2021): 291–96. http://dx.doi.org/10.1007/s40995-021-01239-5.

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Hassan, Mohammad, Muhsin Al Hassan, and Mazen Mostafa. "On Signed Domination of Grid Graph." Open Journal of Discrete Mathematics 10, no. 04 (2020): 96–112. http://dx.doi.org/10.4236/ojdm.2020.104010.

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Ahangar Abdollahzadeh, Hossein, Mustapha Chellali, and Seyed Sheikholeslami. "Signed double roman domination of graphs." Filomat 33, no. 1 (2019): 121–34. http://dx.doi.org/10.2298/fil1901121a.

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红, 霞. "Signed Total Domination Number of Graphs." Advances in Applied Mathematics 07, no. 12 (2018): 1543–48. http://dx.doi.org/10.12677/aam.2018.712180.

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Volkmann, Lutz. "Signed Roman k-domination in Digraphs." Graphs and Combinatorics 32, no. 3 (2015): 1217–27. http://dx.doi.org/10.1007/s00373-015-1641-3.

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Matoušek, Jiří. "On the Signed Domination in Graphs." Combinatorica 20, no. 1 (2000): 103–8. http://dx.doi.org/10.1007/s004930070034.

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Zhao, Yan-cai, Er-fang Shan, Lian-ying Miao, and Zuo-song Liang. "On Signed Star Domination in Graphs." Acta Mathematicae Applicatae Sinica, English Series 35, no. 2 (2019): 452–57. http://dx.doi.org/10.1007/s10255-019-0816-8.

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S, Anandha Prabhavathy. "Complementary Non-negative Signed Domination Number." International Journal of Mathematics Trends and Technology 66, no. 5 (2020): 6–11. http://dx.doi.org/10.14445/22315373/ijmtt-v66i5p502.

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Xu, Baogen. "On signed cycle domination in graphs." Discrete Mathematics 309, no. 4 (2009): 1007–12. http://dx.doi.org/10.1016/j.disc.2008.01.007.

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Volkmann, Lutz. "Signed total Roman domination in graphs." Journal of Combinatorial Optimization 32, no. 3 (2015): 855–71. http://dx.doi.org/10.1007/s10878-015-9906-6.

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Henning, Michael A., and Lutz Volkmann. "Signed Roman k-domination in trees." Discrete Applied Mathematics 186 (May 2015): 98–105. http://dx.doi.org/10.1016/j.dam.2015.01.019.

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Ahangar, Hossein Abdollahzadeh, Mustapha Chellali, and Seyed Mahmoud Sheikholeslami. "Signed double Roman domination in graphs." Discrete Applied Mathematics 257 (March 2019): 1–11. http://dx.doi.org/10.1016/j.dam.2018.09.009.

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Zelinka, Bohdan. "Signed domination numbers of directed graphs." Czechoslovak Mathematical Journal 55, no. 2 (2005): 479–82. http://dx.doi.org/10.1007/s10587-005-0038-5.

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Khoeilar, R., L. Shahbazi, S. M. Sheikholeslami, and Zehui Shao. "Bounds on the signed total Roman 2-domination in graphs." Discrete Mathematics, Algorithms and Applications 12, no. 01 (2020): 2050013. http://dx.doi.org/10.1142/s1793830920500135.

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Streszczenie:

Let [Formula: see text] be an integer and [Formula: see text] be a simple and finite graph with vertex set [Formula: see text]. A signed total Roman [Formula: see text]-dominating function (STR[Formula: see text]DF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (ii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STR[Formula: see text]DF [Formula: see text] is [Formula: see text] and the min

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Hong, Xia, Li Zhang, and Xiaobing Guo. "The Signed Total Mixed Roman Domination Numbers of Graphs." Journal of Physics: Conference Series 2747, no. 1 (2024): 012014. http://dx.doi.org/10.1088/1742-6596/2747/1/012014.

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Abstract The problem of signed domination of graphs is a typical optimization problem. It requires that each vertex and edge be assigned a feasible label so that the sum of assigned labels is minimized. First, we propose the concept of a signed total mixed Roman domination number and introduce the related research progress. Finally, we determine several lower bounds of the signed total mixed Roman domination numbers which depend on the parameters of vertex number and edge number, degree and leaf number.

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