## 1 Introduction  
Distance-hereditary graphs were first introduced and studied in the 1970s as part of the broader exploration of special graph classes characterized by specific structural properties. A graph is defined as distance-hereditary if the distance between every pair of vertices in any connected induced subgraph is the same as in the original graph. This preservation of distances under induced subgraphs has led to distance-hereditary graphs being considered an important and intriguing class within graph theory. The concept was formally defined by [PERSON] in 1977 [1], who provided the initial characterization and properties of distance-hereditary graphs. This work laid the groundwork for further research into the structural and algorithmic aspects of these graphs. The 1980s and 1990s saw significant advancements in the study of distance-hereditary graphs, with researchers exploring various algorithmic applications and characterizations. For instance, [PERSON] and [PERSON] [2] demonstrated that distance-hereditary graphs could be recognized efficiently, leading to a surge in interest in their algorithmic properties. The recognition algorithm for distance-hereditary graphs, which operates in linear time, was a notable milestone, highlighting the practical feasibility of working with these graphs in computational contexts. The study of distance-hereditary graphs continues to be a dynamic and evolving field. One of the critical aspects of distance-hereditary graphs is the relationship with interval functions [3]. The interval function \(I_{G}(u,v)\) in a connected graph captures the set of vertices lying on some shortest \(u,v\)-path, and, in distance-hereditary graphs, every induced path is indeed the shortest. This property has allowed for a novel characterization of these graphs using first-order betweenness axioms and enriching the theoretical framework surrounding them. In addition to these foundational aspects, [PERSON] et al. introduced and explored mutual-visibility in distance-hereditary graphs [4]. Mutual-visibility within a graph per tains to the visibility of vertices along the shortest paths, and it has been shown that the mutual-visibility number can be computed in linear time for distance-hereditary graphs, providing efficient algorithms for specific graph classes. Further expanding on the properties of distance-hereditary graphs, [PERSON] et al. [5] investigated the eccentricity function, demonstrating that it is almost unimodal within these graphs. This research offers a complete characterization of the centers of distance-hereditary graphs and proposes a linear-time algorithm for computing eccentricities, contributing to the efficient processing of these graphs. The complexity of graph problems within distance-hereditary graphs has also been a subject of intense study. For instance, [PERSON] and [PERSON] addressed the fine-grained complexity of the maximum matching problem in graphs of bounded split-width, including distance-hereditary graphs [6]. Their work answers an open question and introduces new tools for the \(b\)-matching problem and dynamic programming. In graph isomorphism, [PERSON] et al. showed that the Weisfeiler-Leman algorithm effectively tests isomorphism for distance-hereditary graphs, reducing the dimension required for this class from the previously known upper bound [7]. This result enhances our ability to differentiate between graphs in this class efficiently. Paired-domination, another key problem in graph theory, has been examined in the context of distance-hereditary graphs [8]. The work provides an \(O(n^{2})\)-time algorithm for determining minimum paired-dominating sets. Moreover, the recognition of distance-hereditary graphs has been explored through distributed interactive proofs [9]. The protocols for recognizing these graphs, with proof sizes as small as \(O(\log n)\) bits, underscore the feasibility of decentralized algorithms in graph theory. Finally, the enumeration of distance-hereditary graphs has been advanced by [PERSON] et al.