Vertex deletion problems on chordal graphs

Abstract

Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yannakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. IntroductionGenerally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. Many classic optimization problems belong to the family of vertex deletion problems, and their algorithms and complexity have been intensively studied. For example, the clique problem and the independent set problem are nothing but the vertex deletion problems to complete graphs and to edgeless graphs respectively. Most interesting graph properties are hereditary: If a graph satisfies this property, then so does every induced subgraph of it. For all the vertex deletion problems to hereditary graph classes, Lewis and Yannakakis [27] have settled their complexity once and for all with a dichotomy result: They are either NP-hard or trivial. Thereafter algorithmic efforts were mostly focused on the nontrivial ones, and the major approaches include approximation algorithms [28], parameterized algorithms [6], and exact algorithms [15].Chordal graphs make one of the most important graph classes. Together with many of its subclasses, it has played important roles in the development of structural graph theory. (We defer their definitions to the next section.) Many algorithms have been developed for vertex deletion problems to chordal graphs and its subclasses,-most notably (unit) interval graphs, cluster graphs, and split graphs; see, e.g., [17,4,10,9,8,34,12,25, 1] for a partial list. After the long progress of algorithmic achievements, some natural questions arise: What is the complexity of transforming a chordal graph to a (unit) interval graph, a cluster graph, a split graph, or a member of some other subclass of chordal graphs? It is quite surprising that this type of problems has not been systematically studied, save few concrete results, e.g., the polynomial-time algorithms for the clique problem, the independent set problem, and the feedback vertex set problem (the object class being forests) [21,33].The same question can be asked for other pair of source and object graph classes. The most important source classes include planar graphs [20,18,16], bipartite graphs [32], and degree-bounded graphs [19]. As one may expect, with special properties imposed on input graphs, the problems become easier, and some of them may not remain NP-hard. Unfortunately, a clear-cut answer to them seems very unlikely, since their complexity would...

On the $d$-Claw Vertex Deletion Problem

“…It is also shown that d-claw-vd remains NP-complete when restricted to split graphs of diameter 2 and to bipartite graphs of diameter 3 (with only two vertices of unbounded degree) and polynomially solvable on bipartite graphs of diameter 2, and thus another dichotomy with respect to diameter. We also define a new class of graphs called d-block graphs which generalize the class of block graphs and show that d-claw-vd is solvable in linear time on d-block graphs, extending the algorithm for cluster-vd on block graphs in [5] to d-claw-vd, and improving the algorithm for (unweighted) 3-claw-vd on block graphs in [2] to 3-block graphs. We note that vertex cover and cluster-vd have been considered by a large number of papers in the context of approximation, exact and parameterized algorithms.…”

“…When restricted to chordal graphs, it is shown in [2] that 3-claw-vd remains NP-complete even on split graphs. The computational complexity of 2-claw-vd on chordal graphs is still unknown [4,5]. Both 2-claw-vd and 3-claw-vd can be solved in polynomial time on block graphs [2,5], a proper subclass of chordal graphs containing all trees.…”

“…The computational complexity of 2-claw-vd on chordal graphs is still unknown [4,5]. Both 2-claw-vd and 3-claw-vd can be solved in polynomial time on block graphs [2,5], a proper subclass of chordal graphs containing all trees. It is well known that the classical NP-complete problem vertex cover remains hard when restricted to planar graphs of maximum degree 3 and arbitrary large girth.…”

“…We also remark that Theorem 6 implies, in particular, that d-clawvd is NP-complete on chordal graphs for any d ≥ 3, while the complexity of 2-claw-vd on chordal graphs is still open (cf. [4,5]). Since it is UGC-hard to approximate min (d − 1)-hvc to a factor (d − 1) − ǫ for any ǫ > 0 [13], Fact 17 implies:…”

Algorithms for deletion problems on split graphs

“…Similarly, in the Split to Threshold Vertex Deletion (STVD) problem the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of at most k vertices such that G − S is a threshold graph. The SBVD and STVD problems were shown to be NP-hard by Cao et al [1]. A split graph G is a block graph if and only if G does not contain an induced diamond, where a diamond is a graph with 4 vertices and 5 edges.…”