Distance-hereditary Graphs Research Articles

αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-Metric Graphs: Radius, Diameter and all Eccentricities

We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-metric (i∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i\\in {\\mathcal {N}}$$\\end{document}) if it satisfies the following αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-metric property for every vertices u, w, v and x: if a shortest path between u and w and a shortest path between x and v share a terminal edge vw, then d(u,x)≥d(u,v)+d(v,x)-i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(u,x)\\ge d(u,v) + d(v,x)-i$$\\end{document}. Roughly, gluing together any two shortest paths along a common terminal edge may not necessarily result in a shortest path but yields a “near-shortest” path with defect at most i. It is known that α0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _0$$\\end{document}-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-metric for i=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1$$\\end{document} and i=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=2$$\\end{document}, respectively. We show that an additive O(i)-approximation of the radius, of the diameter, and in fact of all vertex eccentricities of an αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-metric graph can be computed in total linear time. Our strongest results are obtained for α1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _1$$\\end{document}-metric graphs, for which we prove that a central vertex can be computed in subquadratic time, and even better in linear time for so-called (α1,Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\alpha _1,\\varDelta )$$\\end{document}-metric graphs (a superclass of chordal graphs and of plane triangulations with inner vertices of degree at least 7). The latter answers a question raised in Dragan (Inf Probl Lett 154:105873, 2020), 2020). Our algorithms follow from new results on centers and metric intervals of αi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _i$$\\end{document}-metric graphs. In particular, we prove that the diameter of the center is at most 3i+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3i+2$$\\end{document} (at most 3, if i=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1$$\\end{document}). The latter partly answers a question raised in Yushmanov and Chepoi (Math Probl Cybernet 3:217–232, 1991).

Efficient enumeration of non-isomorphic distance-hereditary graphs and related graphs

We investigate the enumeration problems for the class of distance-hereditary graphs and related graph classes. We first give an enumeration algorithm for the class of distance-hereditary graphs using the recent framework for enumerating every non-isomorphic element in a graph class. In the algorithm, we use the vertex-incremental characterization of distance-hereditary graphs, which consists of three generation rules for adding one vertex to a distance-hereditary graph. Reducing the three generation rules, we can obtain vertex-incremental characterization of cographs and (6,2)-chordal bipartite graphs. These enumeration algorithms are slower than previously known theoretical enumeration algorithms, however, ours are easy to implement. In fact, we implemented our algorithm and obtained catalogs of these graph classes of up to 15 vertices. The class of Ptolemaic graphs is an intersection of the class of chordal graphs and the class of distance-hereditary graphs. We modify the enumeration algorithm for distance-hereditary graphs and obtain an enumeration algorithm for Ptolemaic graphs. The running time of the enumeration of Ptolemaic graphs is much improved from the previously known one, and we also enumerated the graphs up to 15 vertices. We also enumerate the class of 3-leaf power graphs by modifying the algorithm for Ptolemaic graphs. As far as the authors know, some of these graph classes had been counted, however, they had never been enumerated.

Twin-distance-hereditary digraphs

Distance-hereditary graphs are in important graph class in graph theory, as they are well-placed in the graph class hierarchy and permit many algorithmic results. We investigate structural and algorithmic advantages of a directed version of this well-researched graph class. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several twin and pendant vertex operations analogously to undirected distance-hereditary graphs and which still preserves the distance hereditary property. We give a characterization by forbidden induced subdigraphs and place the class in a hierarchy comparing it to related classes. We further show algorithmic advantages concerning directed width parameters, directed graph coloring and some other well-known digraph problems which are NP-hard in general, but computable in polynomial or even linear time on twin-distance-hereditary digraphs. This includes computability of directed path-width and tree-width in linear time and the directed chromatic number in polynomial time. From our result that directed twin-distance-hereditary graphs have directed clique-width at most 3 it follows by Courcelle's theorem on directed clique-width that we can compute every graph problem describable in monadic second-order logic on quantification over vertices and vertex sets as well as some further problems like Hamiltonian Path/Cycle in polynomial time.

Mutual-visibility in distance-hereditary graphs: a linear-time algorithm

The concept of mutual-visibility in graphs has been recently introduced. If X is a subset of vertices of a graph G, then vertices u and v are X-visible if there exists a shortest u, v-path P such that V(P) ∩ X⊆ {u, v}. If every two vertices from X are X-visible, then X is a mutual-visibility set. The mutual-visibility number of G is the cardinality of a largest mutual-visibility set of G. It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class.

Composition and product of cover-incomparability graphs

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs from posets. C-I graphs are identified among chordal graphs, distance-hereditary graphs, Ptolemaic graphs, split graphs, threshold graphs, bisplit graphs, block graphs and cographs. Thus only a few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.

Restrained condition on double Roman dominating functions

We continue the study of restrained double Roman domination in graphs. For a graph G=(V(G),E(G)), a double Roman dominating function f is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by {v∈V(G)∣f(v)=0} has no isolated vertices. The restrained double Roman domination number (RDRD number) γrdR(G) is the minimum weight ∑v∈V(G)f(v) taken over all RDRD functions of G.We first prove that the problem of computing γrdR is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between γrdR and some well-known parameters such as restrained domination number γr, domination number γ or restrained Roman domination number γrR are investigated in this paper by bounding γrdR from below and above involving γr, γ or γrR for general graphs, respectively. We prove that γrdR(T)≥n+2 for any tree T≠K1,n−1 of order n≥2 and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper.

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in \(\tilde{\mathcal{O}}(f(k) \cdot (n+m))\) time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and \(P_4\)-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle’s theorem to this problem directly, because a matching cannot be expressed in \(MSO_1\) logic. Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. We also present how to compute the Edmonds-Gallai decomposition in almost linear time by using the same framework. For that, we do apply Courcelle’s theorem but to the classic Tutte-Berge formula, that can easily be expressed as a \(CMSO_1\) optimization problem. Doing so, we can compute the cardinality of a maximum matching, but not the matching itself. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs and with a distributed version of Courcelle’s theorem (Courcelle and Vanicat, DAM 2016) – of which we present here a slightly stronger version than the standard one in the literature. Finally, for the bipartite graphs of clique-width at most k, we present an alternative \(\tilde{\mathcal{O}}(k^2\cdot (n+m))\)-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC’21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.

Injective Hulls of Various Graph Classes

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph \(\mathcal {H}(G)\) into which G isometrically embeds; \(\mathcal {H}(G)\) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs \(\mathcal {C}\) is closed under Hellification if \(G \in \mathcal {C}\) implies \(\mathcal {H}(G) \in \mathcal {C}\). We identify several graph classes that are closed under Hellification. We show that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computable injective hull are of particular interest. A linear-time algorithm to construct the injective hull of any distance-hereditary graph is provided and we show that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute in subexponential time. In particular, there are split graphs, cocomparability graphs, and bipartite graphs G such that \(\mathcal {H}(G)\) contains \(\Omega (a^{n})\) vertices, where \(n=|V(G)|\) and \(a>1\).

Towards Constant-Factor Approximation for Chordal/Distance-Hereditary Vertex Deletion

For a family of graphs \(\mathcal {F}\), Weighted \(\mathcal {F}\)-Deletion is the problem for which the input is a vertex weighted graph \(G = (V, E)\) and the goal is to delete \(S \subseteq V\) with minimum weight such that \(G \setminus S \in \mathcal {F}\). Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when \(\mathcal {F}\) is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

A necessary condition for the equality of the clique number and the convexity number of a graph

Let G be a graph, u and v two vertices of G and X a subset of V(G). A u−vgeodesic is a path between u and v of minimum length. Ig(u,v) is the set of vertices that lie on any u−v geodesic and Ig(X) is the set ⋃u,v∈XIg(u,v). X is g-convex if Ig(X)=X. The convexity number, con(G), of G is the maximum cardinality of a proper g-convex set of G. The clique number, ω(G), of G is the maximum cardinality of a clique of G. If G is a connected not complete graph then ω(G)≤con(G). In this paper a necessary condition for ω(G)=con(G) is provided and, on the basis of this condition, both the class of distance-hereditary graphs for which ω(G)=con(G) and the class of chordal graphs for which ω(G)=con(G) are characterized.

A tight relation between series-parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum value.

A divide-and-conquer approach for reconstruction of {C≥5}-free graphs via betweenness queries

We study the query complexity of reconstructing {C≥5}-free graphs with respect to the betweenness oracle. In particular, we show that hidden {C≥5}-free graphs can be reconstructed by using O(Δ14⋅log2⁡n+Δ9⋅nlog2⁡n) betweenness queries in expectation, where Δ denotes the maximum degree of the given graph and n denotes the number of vertices. In addition, we propose two improved randomized algorithms for two subclasses of {C≥5}-free graphs, namely the distance-hereditary graphs and the chordal graphs. For the former class, our algorithm uses O(Δ10⋅log2⁡n+Δ5⋅nlog2⁡n) betweenness queries in expectation, and for the latter class, our algorithm uses O(Δ2⋅nlog2⁡n) betweenness queries in expectation.

A note on the triameter of graphs

In this article, we answer three questions from the paper (Das, 2021). We observe a tight lower bound for the triameter of trees in terms of order and number of leaves. We show that in a connected block graph any triametral triple of vertices contains a diametral pair and that any diametral pair of vertices can be extended to a triametral triple. We also present several open problems concerning the interplay between triametral triples, diametral pairs and peripheral vertices in median and distance-hereditary graphs.

Computing a metric basis of a bipartite distance-hereditary graph

A vertex x of a connected graph G resolves two distinct vertices u and v in V(G) if the distance between u and x differs from the distance between v and x. A subset X of V(G)resolves two distinct vertices u and v in G if there exists a vertex x in X that resolves u and v; X is a metric generator of G if, for every pair of distinct vertices u and v of G, X resolves u and v and is a metric basis of G if X is a metric generator of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis of G. The problem of finding the metric dimension of an arbitrary graph is NP-hard. In this paper we show that the problem is solvable in linear time for the class of bipartite distance-hereditary graphs.

Maximal double Roman domination in graphs

A maximal double Roman dominating function (MDRDF) on a graph G=(V,E) is a function f:V(G)→{0,1,2,3} such that (i) every vertex v with f(v)=0 is adjacent to least two vertices assigned 2 or to at least one vertex assigned 3, (ii) every vertex v with f(v)=1 is adjacent to at least one vertex assigned 2 or 3 and (iii) the set {w∈V|f(w)=0} is not a dominating set of G. The weight of a MDRDF is the sum of its function values over all vertices, and the maximal double Roman domination number γdRm(G) is the minimum weight of an MDRDF on G. In this paper, we initiate the study of maximal double Roman domination. We first show that the problem of determining γdRm(G) is NP-complete for bipartite, chordal and planar graphs. But it is solvable in linear time for bounded clique-width graphs including trees, cographs and distance-hereditary graphs. Moreover, we establish various relationships relating γdRm(G) to some domination parameters. For the class of trees, we show that for every tree T of order n≥4,γdRm(T)≤54n and we characterize all trees attaining the bound. Finally, the exact values of γdRm(G) are given for paths and cycles.

The [formula omitted]-Matching problem in distance-hereditary graphs and beyond

We make progress on the fine-grained complexity of Maximum-Cardinality Matching within graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA’17) and graphs of bounded modular-width (Coudert et al., SODA’18). We present such algorithm for bounded split-width graphs — a broad generalization of graphs of bounded modular-width, of which an interesting subclass is the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((klog2k)⋅(m+n)⋅logn)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we answer an open question of Coudert et al. (SODA’18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

Getting new algorithmic results by extending distance-hereditary graphs via split composition.

In this paper, we consider the graph class denoted as Gen(∗;P3,C3,C5). It contains all graphs that can be generated by the split composition operation using path P3, cycle C3, and any cycle C5 as components. This graph class extends the well-known class of distance-hereditary graphs, which corresponds, according to the adopted generative notation, to Gen(∗;P3,C3). We also use the concept of stretch number for providing a relationship between Gen(∗;P3,C3) and the hierarchy formed by the graph classes DH(k), with k ≥1, where DH(1) also coincides with the class of distance-hereditary graphs. For the addressed graph class, we prove there exist efficient algorithms for several basic combinatorial problems, like recognition, stretch number, stability number, clique number, domination number, chromatic number, and graph isomorphism. We also prove that graphs in the new class have bounded clique-width.

On the complexity of minimum maximal uniquely restricted matching

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. A matching M in a graph G is called a uniquely restricted matching if, G[V(M)], the subgraph of G induced by the set of M-saturated vertices of G contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching in G of minimum cardinality and Decide-Min-Max-UR Matching problem, the decision version of this problem takes a graph G and an integer k and asks whether G admits a maximal uniquely restricted matching of cardinality at most k. It is known that the Decide-Min-Max-UR Matching problem is NP-complete. In this paper, we strengthen this result by proving that the Decide-Min-Max-UR Matching problem remains NP-complete for chordal bipartite graphs, star-convex bipartite graphs, chordal graphs, and doubly chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite distance-hereditary graphs and linear time solvable for bipartite permutation graphs, proper interval graphs, and threshold graphs. Finally, we prove that the Min-Max-UR Matching problem is APX-complete for graphs with maximum degree 4.

A Quasi-Hole Detection Algorithm for Recognizing k-Distance-Hereditary Graphs, with k < 2

Cicerone and Di Stefano defined and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the k-distance-hereditary graphs such that k<2. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.

A Polynomial Kernel for Distance-Hereditary Vertex Deletion

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph G on n vertices and an integer k, whether there is a set S of at most k vertices in G such that $$G-S$$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width because distance-hereditary graphs are exactly the graphs of rank-width at most 1. Eiben, Ganian, and Kwon (JCSS’ 18) proved that Distance-Hereditary Vertex Deletion can be solved in time $$2^{{\mathcal {O}}(k)}n^{{\mathcal {O}}(1)}$$ , and asked whether it admits a polynomial kernelization. We show that this problem admits a polynomial kernel, answering this question positively. For this, we use a similar idea for obtaining an approximate solution for Chordal Vertex Deletion due to Jansen and Pilipczuk (SIDMA’ 18) to obtain an approximate solution with $${\mathcal {O}}(k^3\log n+ k^2\log ^2 n)$$ vertices when the problem is a Yes-instance, and we exploit the structure of split decompositions of distance-hereditary graphs to reduce the total size.