Undirected Path Graphs Research Articles

Injective coloring of subclasses of chordal graphs

An injective k-coloring of a graph G=(V,E) is a function f:V→{1,2,…,k} such that for every pair of vertices u and v having a common neighbor, f(u)≠f(v). The injective chromatic number χi(G) of a graph G is the minimum k for which G admits an injective k-coloring. Given a graph G and a positive integer k, Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. Decide Injective Coloring Problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this result by proving that Decide Injective coloring Problem remains NP-complete for undirected path graphs, a proper subclass of chordal graphs. Moreover, we show that it is not possible to approximate the injective chromatic number of an undirected path graph within a factor of n1/3−ε in polynomial time for every ε>0 unless ZPP = NP. On the positive side, we prove that the injective chromatic number of an interval graph is either Δ(G) or Δ(G)+1, where Δ(G) is the maximum degree of G. We also characterize the interval graphs having χi(G)=Δ(G) and χi(G)=Δ(G)+1. As a consequence of this characterization, we obtain a linear time algorithm to find the injective chromatic number of an interval graph.

Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

AbstractChordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

Algorithmic Aspects of Vertex-edge Domination in Some Graphs

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set \(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document} and a set S⊆V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\subseteq V$$\\end{document}, we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains np-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider the leafage, which measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, we provide an algorithm for solving SFVS with running time nO(ℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{O(\\ell )}$$\\end{document}, thus improving upon nO(ℓ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{O(\\ell ^2)}$$\\end{document}, which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w[1]-hard parameterized by ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}. Pushing further our positive result, it is natural to also consider the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains np-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

Algorithmic results in Roman dominating functions on graphs

Given a graph G=(V,E), a subset D⊆V (respectively, function f:V→{0,1,2}) is a dominating set (DS) (respectively, Roman dominating function (RDF)) of G if each vertex v∈V∖D (respectively, v∈V with f(v)=0) is adjacent to a vertex u∈D (respectively, u∈V with f(u)=2). The domination number of G is the minimum cardinality of an DS of G and the Roman domination number of G is the minimum weight of an RDF f of G, where the weight of f is ∑v∈Vf(v). The (Roman) domination problem is to compute the (Roman) domination number of a given graph. In this paper, we study the Roman domination problem. We show that the complexity of the problem differs from the complexity of the domination problem and the problem is NP-complete for circle graphs and undirected path graphs and is APX-complete for graphs of degree at most 4. We also propose an integer linear programming (ILP) formulation with polynomial number of constraints for the problem. Additionally, we use the ILP formulation to give an H(Δ(G)+1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G. Furthermore, we show that the optimization version of the problem on split and chordal graphs cannot be approximated in polynomial time within (1/2−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)).

Revising Johnson’s table for the 21st century

What does it mean today to study a problem from a computational point of view? We focus on parameterized complexity and on Column 16 “Graph Restrictions and Their Effect” of D.S. Johnson’s Ongoing guide, where several puzzles were proposed in a summary table with 30 graph classes as rows and 11 problems as columns. Several of the 330 entries remain unclassified into Polynomial or NP-complete after 35 years. We provide a full dichotomy for the Steiner Tree column by proving that the problem is NP-complete when restricted to Undirected Path graphs. We revise Johnson’s summary table according to the granularity provided by the parameterized complexity for NP-complete problems.

Exact square coloring of certain classes of graphs: Complexity and algorithms

A vertex coloring of a graph G=(V,E) is called an exact square coloring of G if any two vertices at distance exactly 2 receive different colors. The minimum number of colors required by an exact square coloring is called the exact square chromatic number of G and is denoted by χ[#2](G). Given a graph G and a positive integer k, the Exact Square Coloring Problem is to decide whether G admits an exact square coloring using k colors. It is known that Exact Square Coloring Problem is NP-complete for chordal graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for undirected path graphs, which is a proper subclass of chordal graphs. However, we give linear time algorithms for computing the exact square chromatic number of proper interval graphs and threshold graphs, which are proper subclasses of chordal graphs. Moreover, for a proper interval graph G, we show that χ[#2](G)≤3. We also propose a polynomial time algorithm to produce an exact square coloring of a block graph G using at most χ[#2](G)+1 colors. Next, we study a lower bound of χ[#2](G). A subset S of vertices of a graph G=(V,E) is called an exact square clique of G if the distance between any two vertices in S is exactly 2. The cardinality of the maximum exact square clique of G is called the exact square clique number of G and is denoted by ω[#2](G). Clearly, ω[#2](G)≤χ[#2](G). Given a graph G and a positive integer k, the problem of deciding whether ω[#2](G) is at least k, is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we strengthen these results by proving that this problem remains NP-complete for undirected path graphs, perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs. We also compute the exact value of ω[#2](G) for proper interval graphs, threshold graphs, block graphs and convex bipartite graphs.

Total 2-Rainbow Domination in Graphs

A total k-rainbow dominating function on a graph G=(V,E) is a function f:V(G)→2{1,2,…,k} such that (i) ∪u∈N(v)f(u)={1,2,…,k} for every vertex v with f(v)=∅, (ii) ∪u∈N(v)f(u)≠∅ for f(v)≠∅. The weight of a total 2-rainbow dominating function is denoted by ω(f)=∑v∈V(G)|f(v)|. The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G. The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem.

Roman {k}-domination in trees and complexity results for some classes of graphs

In this paper, we study Roman {k}-dominating functions on a graph G with vertex set V for a positive integer k: a variant of {k}-dominating functions, generations of Roman $$\{2\}$$ -dominating functions and the characteristic functions of dominating sets, respectively, which unify classic domination parameters with certain Roman domination parameters on G. Let $$k\ge 1$$ be an integer, and a function $$f:V \rightarrow \{0,1,\dots ,k\}$$ defined on V called a Roman $$\{k\}$$ -dominating function if for every vertex $$v\in V$$ with $$f(v)=0$$ , $$\sum _{u\in N(v)}f(u)\ge k$$ , where N(v) is the open neighborhood of v in G. The minimum value $$\sum _{u\in V}f(u)$$ for a Roman $$\{k\}$$ -dominating function f on G is called the Roman $$\{k\}$$ -domination number of G, denoted by $$\gamma _{\{Rk\}}(G)$$ . We first present bounds on $$\gamma _{\{Rk\}}(G)$$ in terms of other domination parameters, including $$\gamma _{\{Rk\}}(G)\le k\gamma (G)$$ . Secondly, we show one of our main results: characterizing the trees achieving equality in the bound mentioned above, which generalizes M.A. Henning and W.F. klostermeyer’s results on the Roman {2}-domination number (Henning and Klostermeyer in Discrete Appl Math 217:557–564, 2017). Finally, we show that for every fixed $$k\in \mathbb {Z_{+}}$$ , associated decision problem for the Roman $$\{k\}$$ -domination is NP-complete, even for bipartite planar graphs, chordal bipartite graphs and undirected path graphs.

A story of diameter, radius, and (almost) Helly property

AbstractWe present new algorithmic results for the class of Helly graphs, that is, for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph that makes of the latter an important class of graphs in metric graph theory. We study diameter and radius computations within the Helly graphs, and related graph classes. This is in part motivated by a conjecture on the fine‐grained complexity of these two distance problems within the graph classes of bounded fractional Helly number—that contain as particular cases the proper minor‐closed graph classes and the bounded clique‐width graphs. Note that under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, we first present algorithms which given an n‐vertex m‐edge Helly graph G as input, compute with high probability (w.h.p.) its radius and its diameter in time (i.e., subquadratic in n + m). Our algorithms are based on the Helly property and on the unimodality of the eccentricity function in Helly graphs: every vertex of locally minimum eccentricity is a central vertex. Then, we improve our results for the C4‐free Helly graphs, that are exactly the Helly graphs whose balls are convex. For this subclass, we present linear‐time algorithms for computing the eccentricity of all vertices. Doing so, we generalize previous results on strongly chordal graphs to a much larger subclass, that includes, among others, all the bridged Helly graphs and the hereditary Helly graphs. Lastly, we derive approximate versions of our results for the class of chordal graphs: with the latter satisfying an almost‐Helly‐type property, and a stronger (induced‐path) convexity property than the C4‐free Helly graphs. For the chordal graphs, we can compute in quasi linear time the eccentricity of all vertices with an additive one‐sided error of at most one, which is best possible under the strong exponential‐time hypothesis. This answers an open question of Dragan. In fact, we obtain this last result as a byproduct from a more general reduction: from diameter computation on chordal graphs to the Disjoint Sets problem. Roughly, it implies that the split graphs are the only hard instances for diameter computation on chordal graphs. We also get from our reduction that on any subclass of chordal graphs with constant VC‐dimension (and so, for undirected path graphs), the diameter can be computed in truly subquadratic time.

Algorithmic aspects of k-part degree restricted domination in graphs

The concept of network is predominantly used in several applications of computer communication networks. It is also a fact that the dominating set acts as a virtual backbone in a communication network. These networks are vulnerable to breakdown due to various causes, including traffic congestion. In such an environment, it is necessary to regulate the traffic so that these vulnerabilities could be reasonably controlled. Motivated by this, [Formula: see text]-part degree restricted domination is defined as follows. For a positive integer [Formula: see text], a dominating set [Formula: see text] of a graph [Formula: see text] is said to be a [Formula: see text]-part degree restricted dominating set ([Formula: see text]-DRD set) if for all [Formula: see text], there exists a set [Formula: see text] such that [Formula: see text] and [Formula: see text]. The minimum cardinality of a [Formula: see text]-DRD set of a graph [Formula: see text] is called the [Formula: see text]-part degree restricted domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we present a polynomial time reduction that proves the NP -completeness of the [Formula: see text]-part degree restricted domination problem for bipartite graphs, chordal graphs, undirected path graphs, chordal bipartite graphs, circle graphs, planar graphs and split graphs. We propose a polynomial time algorithm to compute a minimum [Formula: see text]-DRD set of a tree and minimal [Formula: see text]-DRD set of a graph.

Algorithm and hardness results on neighborhood total domination in graphs

A set D⊆V of a graph G=(V,E) is called a neighborhood total dominating set of G if D is a dominating set and the subgraph of G induced by the open neighborhood of D has no isolated vertex. Given a graph G, Min-NTDS is the problem of finding a neighborhood total dominating set of G of minimum cardinality. The decision version of Min-NTDS is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we extend this NP-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs. We show that for a given graph G=(V,E), Min-NTDS cannot be approximated within a factor of (1−ε)log⁡|V|, unless NP⊆DTIME(|V|O(log⁡log⁡|V|)) and can be approximated within a factor of O(log⁡Δ), where Δ is the maximum degree of the graph G. Finally, we show that Min-NTDS is APX-complete for graphs of degree at most 3.

Algorithmic results on double Roman domination in graphs

Given a graph $$G=(V,E)$$, a function $$f:V\longrightarrow \{0,1,2,3\}$$ is called a double Roman dominating function on G if (i) for every $$v\in V$$ with $$f(v)=0$$, there are at least two neighbors of v that are assigned 2 under f or at least a neighbor of v that is assigned 3 under f, and (ii) for every vertex v with $$f(v)=1$$, there is at least one neighbor w of v with $$f(w)\ge 2$$. The weight of a double Roman dominating function f is $$f(V)=\sum _{u\in V}f(u)$$. The double Roman domination number of G, denoted by $$\gamma _{dR}(G)$$ is the minimum weight of a double Roman dominating function on G. For a graph $$G=(V,E)$$, Min-Double-RDF is to find a double Roman dominating function f with $$f(V)=\gamma _{dR}(G)$$. The decision version of Min-Double-RDF is shown to be NP-complete for chordal graphs and bipartite graphs. In this paper, we first strengthen the known NP-completeness of the decision version of Min-Double-RDF by showing that the decision version of Min-Double-RDF remains NP-complete for undirected path graphs, chordal bipartite graphs, and circle graphs. We then present linear time algorithms for computing the double Roman domination number in proper interval graphs and block graphs. We then discuss on the approximability of Min-Double-RDF and present a 2-approximation algorithm in 3-regular bipartite graphs.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, respectively total k-dominating set, in a given graph, are referred to as k-domination, respectively total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from previous works by Bui-Xuan et al. (2013) [8] and by Belmonte and Vatshelle (2013) [3] that these two families of problems are solvable in time O(|V(G)|3k+4) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V(G)|2k) nodes and O(|V(G)|4k) arcs. We show that a suitable implementation, which avoids constructing all arcs of the digraph, leads to a running time of O(|V(G)|3k). The algorithms are also applicable to the weighted case.

On computing a minimum secure dominating set in block graphs

In a graph $$G=(V,E)$$G=(V,E), a set $$D \subseteq V$$D⊆V is said to be a dominating set of G if for every vertex $$u\in V{\setminus }D$$u?V\D, there exists a vertex $$v\in D$$v?D such that $$uv\in E$$uv?E. A secure dominating set of the graph G is a dominating set D of G such that for every $$u\in V{\setminus }D$$u?V\D, there exists a vertex $$v\in D$$v?D such that $$uv\in E$$uv?E and $$(D{\setminus }\{v\})\cup \{u\}$$(D\{v})?{u} is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.

A linear time algorithm to compute a minimum restrained dominating set in proper interval graphs

A set D ⊆ V is a restrained dominating set of a graph G = (V, E) if every vertex in V\D is adjacent to a vertex in D and a vertex in V\D. Given a graph G and a positive integer k, the restrained domination problem is to check whether G has a restrained dominating set of size at most k. The restrained domination problem is known to be NP-complete even for chordal graphs. In this paper, we propose a linear time algorithm to compute a minimum restrained dominating set of a proper interval graph. We present a polynomial time reduction that proves the NP-completeness of the restrained domination problem for undirected path graphs, chordal bipartite graphs, circle graphs, and planar graphs.

On the algorithmic complexity of edge total domination

Let G=(V,E) be a graph with vertex set V and edge set E. A subset F⊆E is an edge total dominating set if every edge e∈E is adjacent to at least one edge in F. The edge total domination problem is to find a minimum edge total dominating set of G. In the present paper, we prove that the edge total domination problem is NP-complete for planar graphs with maximum degree three, and for undirected path graphs, a subclass of chordal graphs and a superclass of trees. We also design a linear-time algorithm for solving this problem in a tree.

On the algorithmic complexity of [formula omitted]-tuple total domination

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a set D⊆V(G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for the k-tuple total domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph, which include trees, block graphs, cacti and block-cactus graphs. We also establish NP-hardness of the k-tuple total domination problem in undirected path graphs.

A linear time algorithm for liarʼs domination problem in proper interval graphs

Let G=(V,E) be a graph without isolated vertices and having at least 3 vertices. A set L⊆V(G) is a liarʼs dominating set if (1) |NG[v]∩L|⩾2 for all v∈V(G), and (2) |(NG[u]∪NG[v])∩L|⩾3 for every pair u,v∈V(G) of distinct vertices in G, where NG[x]={y∈V|xy∈E}∪{x} is the closed neighborhood of x in G. Given a graph G and a positive integer k, the liarʼs domination problem is to check whether G has a liarʼs dominating set of size at most k. The liarʼs domination problem is known to be NP-complete for general graphs. In this paper, we propose a linear time algorithm for computing a minimum cardinality liarʼs dominating set in a proper interval graph. We also strengthen the NP-completeness result of liarʼs domination problem for general graphs by proving that the problem remains NP-complete even for undirected path graphs which is a super class of proper interval graphs.

Computing a minimum outer-connected dominating set for the class of chordal graphs

For a graph G=(V,E), a dominating set is a set D⊆V such that every vertex v∈V∖D has a neighbor in D. Given a graph G=(V,E) and a positive integer k, the minimum outer-connected dominating set problem for G is to decide whether G has a dominating set D of cardinality at most k such that G[V∖D], the induced subgraph by G on V∖D, is connected. In this paper, we consider the complexity of the minimum outer-connected dominating set problem for the class of chordal graphs. In particular, we show that the minimum outer-connected dominating set problem is NP-complete for doubly chordal graphs and undirected path graphs, two well studied subclasses of chordal graphs. We also give a linear time algorithm for computing a minimum outer-connected dominating set in proper interval graphs. Notice that proper interval graphs form a subclass of undirected path graphs as well as doubly chordal graphs.