## 1 Introduction
[10], who developed algorithms that are not only theoretically sound but also practical to implement. These algorithms have led to the creation of catalogs for graph classes up to 15 vertices, demonstrating the practical utility of these methods. The clique transversal problem is a fundamental concept in graph theory, involving the identification of a minimum subset of vertices that intersects all maximal cliques in a graph. This problem, along with its variations--such as the \(k\)-fold clique, \(\{k\}\)-clique, minus clique, and signed clique transversal problems--has attracted significant attention due to its practical applications [11; 12; 13; 14] and theoretical significance [15; 16; 17; 18]. The \(k\)-fold clique, \(\{k\}\)-clique, minus clique, and signed clique transversal problems remain unexplored challenges for distance-hereditary graphs. This paper aims to investigate the four clique transversal problems on distance-hereditary graphs. By focusing on these specific problems, this study aims to deepen the understanding of the computational challenges associated with them and to explore the potential for developing efficient algorithms to solve them. The insights gained from this research are expected to contribute significantly to the broader field of graph theory and to provide practical tools for applications in various domains where graph-based modeling is essential. To solve the four clique transversal problems, this paper identifies their common characteristics and introduces the \((p,d,\ell,b)\)-clique transversal problem. It demonstrates that the \(k\)-fold clique, \(k\)-clique, minus clique, and signed clique transversal problems are all special cases of the \((p,d,\ell,b)\)-clique transversal problem. This formulation provides a unified approach to solving these four problems. We then present a _dynamic programming_ algorithm specifically designed to solve the \((p,d,\ell,b)\)-clique transversal problem. Dynamic programming is a versatile and powerful technique for solving complex problems by breaking them down into smaller subproblems, solving all the smaller subproblems, and storing their results, and then combining their solutions to solve the original problem. It is particularly effective for optimization problems, where the goal is to find the best solution among many possible options. The key idea behind dynamic programming is to store the results of subproblems and reuse them when needed, thereby avoiding redundant calculations and significantly improving efficiency. The development of a dynamic programming algorithm typically involves the following steps:  
1. Characterize the structure of an optimal solution: Identify how the optimal solution to the problem can be composed of the solutions to its subproblems. This step involves understanding the relationship between the original problem and its subproblems and how to combine the subproblem solutions to form the optimal solution for the entire problem;
2. Define the recursive formula: Once the structure of the optimal solution is understood, the next step is to define the recursive relationship (or recurrence relation) that expresses the solution to the original problem in terms of the solutions to its subproblems. This formula is central to the dynamic programming approach, as it dictates how the subproblem solutions are combined;
3. Compute the value of an optimal solution in a bottom-up fashion using the recursive formula;
4. Analyze the time complexity: The time complexity is generally proportional to the number of subproblems multiplied by the time required to solve each subproblem. For a fundamental understanding of dynamic programming algorithms and their examples, readers are encouraged to consult _Introduction to Algorithms_[19].