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On dynamic monopolies of graphs with general thresholds
Combinatorics · Submitted 6 Mar 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds.
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The abstract, sentence by sentence
Let $G$ be a graph and ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be dynamic monopoly (or simply dynamo) if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i=1,..., k-1$ each vertex $v$ in $D_{i+1}$ has at least $t(v)$ neighbors in $D_0\cup ...\cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. We denote the smallest size of any dynamic monopoly of $G$, with a given threshold assignment, by $dyn(G)$. In this paper we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. Next we introduce dynamo-unbounded families of graphs and prove some related results. We also define the concept of a homogenious society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G$ be a graph and ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. | background 1.00 | 0.09 | 0.01 | asserted 0.75 |
| s1 | A subset of vertices $D$ is said to be dynamic monopoly (or simply dynamo) if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i=1,..., k-1$ each vertex $v$ in $D_{i+1}$ has at least $t(v)$ neighbors in $D_0\cup ...\cup D_i$. | background 0.99 | 0.09 | 0.03 | asserted 0.99 |
| s2 | Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. | background 0.91 | 0.08 | 0.03 | asserted 0.99 |
| s3 | We denote the smallest size of any dynamic monopoly of $G$, with a given threshold assignment, by $dyn(G)$. | background 0.90 | 0.31 | 0.03 | asserted 0.98 |
| s4 | In this paper we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. | contribution 0.99 | 0.93 | 0.05 | asserted 0.99 |
| s5 | Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. | contribution 0.71 | 0.96 | 0.21 | asserted 1.00 |
| s6 | Next we introduce dynamo-unbounded families of graphs and prove some related results. | contribution 1.00 | 0.95 | 0.06 | asserted 0.99 |
| s7 | We also define the concept of a homogenious society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. | contribution 0.97 | 0.94 | 0.08 | asserted 1.00 |
| s8 | Finally we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases. | contribution 0.87 | 0.95 | 0.11 | asserted 1.00 |
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Original abstract
Let $G$ be a graph and ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be dynamic monopoly (or simply dynamo) if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i=1,..., k-1$ each vertex $v$ in $D_{i+1}$ has at least $t(v)$ neighbors in $D_0\cup ...\cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. We denote the smallest size of any dynamic monopoly of $G$, with a given threshold assignment, by $dyn(G)$. In this paper we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. Next we introduce dynamo-unbounded families of graphs and prove some related results. We also define the concept of a homogenious society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases.
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