## 4 Results and Discussion
### Watermark Robustness
#### 4.2.2 Resistance to Vertex Attack  
The \(NC\) values for both non-feature vertex zero-watermarks of ZC and SZ are 1 under different levels of vertex addition attacks (Figure 6a), and the corresponding trend lines for ZC/SVD and SZ/SVD remain straight at \(NC=1\). The feature vertex zero-watermarks have \(NC\) values greater than 0.8 for attacks under the intensity of the first 40% and the first 50% for ZC and SZ, respectively. The SZ/DTM curve is higher than the ZC/DTM curve. The watermark qualities gradually reduce with increased attack intensity. The \(NC\) values for the dual zero-watermarking scheme are always 1 under  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Cropping** & **ZC/DTM** & **ZC/SVD** & **SZ/DTM** & **SZ/SVD** & \begin{tabular}{c} **The** \\ **Proposed** \\ \end{tabular} & \
\begin{tabular}{c} **[PERSON] et al.** \\ **(2002)** [1]** \\ \end{tabular} &
\begin{tabular}{c} **[PERSON] et al.** \\ **(2003)** [24]** \\ \end{tabular} \\ \hline
**Pattern 1** & 0 & 45.0 & 0 & 43.5 & 0 & 1.4 & 0.0 & 0.0 \\ \hline Pattern 2 & 0 & 43.6 & 0 & 44.6 & 0 & 1.8 & 0.1 & 0.4 \\ Pattern 3 & 0 & 62.6 & 0 & 65.0 & 0 & 7.0 & 0.7 & 1.3 \\ Pattern 4 & 0 & 67.0 & 0 & 63.6 & 0 & 6.8 & 0.4 & 0.0 \\ Pattern 5 & 0 different levels of vertex addition attacks, which are the same as the trend of the SVD-based watermark. As seen from Figure 6b, the feature vertex watermark, the non-feature vertex watermark and the dual zero-watermark all show to be robust under the vertex deletion attack. The watermarks based on feature vertex show a slow downward trend for both the ZC and SZ datasets and have the _NC_ values greater than 0.8 for the deletion intensity up to 55%. The watermarks based on the non-feature vertex perform better than those based on the feature vertex given that their _NC_ values keep above 0.8 with a slower declining trend than those of ZC/DTM and SZ/DTM. The dual zero-watermarks combine the strengths of watermarks constructed based on both feature and non-feature vertex and have the same trends as the SVD-based watermark that shows robust to vertex deletion.  
In the process of extraction, we find that when implementing feature vertex extraction by the Douglas-Peucker algorithm, the random increased vertices cannot be compressed, but as the feature vertices to build DTM, so vertex addition attack has no effect on the non-feature vertex watermark. The feature vertex watermark based on the abundant feature information and local stability, early insertion of fewer vertices had little effect on it. When the increasing strength of the attack affects the stability of all local watermarks, the NC values of feature vertex watermark decreased greatly. The feature vertex watermark in SZ is better than it is in ZC, which benefits from the larger feature information in SZ to increase the uncertainty of vertex addition attack, so the local watermarks in SZ are superior. In the proposed scheme, the number of feature vertices is much fewer than non-feature vertices, which determines the randomly deleted feature vertices are very few, so the DTM-based feature vertex watermark shows good robustness against vertex deletion attacks. Reliant on the stability of singular values and the characteristic of scheme, the non-feature vertex watermark also shows good robustness under vertex deletion, even obviously better than the feature vertex watermark. The optimal watermark extracted from the dual zero-watermark is almost the same as the non-feature vertex watermark, so it inherits the robustness advantage of the non-feature vertex watermark in the vertex attack.