Planar Chordal Graphs Research Articles

Algorithm and hardness results on hop domination in graphs

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G=(V,E) is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. Given a graph G, Min-HDS is the problem of finding a hop dominating set of G of minimum cardinality. The decision version of Min-HDS is known to be NP-complete for planar bipartite graphs and planar chordal graphs, and hence for bipartite graphs and chordal graphs. In this paper, we present a linear time algorithm for computing a minimum hop dominating set in bipartite permutation graphs, which is a subclass of bipartite graphs. We also show that Min-HDS cannot be approximated within a factor of (1−ε)ln⁡|V|, unless P=NP and can be approximated within a factor of 1+ln⁡(Δ(Δ−1)+1), where Δ denotes the maximum degree in the graph G. Finally, we show that Min-HDS is APX-complete for bipartite graphs of maximum degree 3.

On the complexity of some hop domination parameters

A hop Roman dominating function (HRDF) on a graph G = ( V , E ) is a function f : V → {0, 1, 2} having the property that for every vertex v ∈ V with f ( v ) = 0 there is a vertex u with f ( u ) = 2 and d ( u , v ) = 2 . The weight of an HRDF f is the sum of its values on V . The minimum weight of an HRDF on G is called the hop Roman domination number of G . An HRDF f is a hop Roman independent dominating function (HRIDF) if for any pair v , w with f ( v ) > 0 and f ( w ) > 0 , d ( v , w ) ≠ 2 . The minimum weight of an HRIDF on G is called the hop Roman independent domination number of G . In this paper, we study the complexity of the hop independent dominating problem, the hop Roman domination function problem and the hop Roman independent domination function problem, and show that the decision problem for each of the above three problems is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.

A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected subgraph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an analogous result for the case when the input graph is a maximal outerplanar graph.

Graphs whose flow polynomials have only integral roots

We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.

One‐phase algorithm for the determination of minimal vertex separators of chordal graphs

AbstractThe set of minimal vertex separators of chordal graphs is usually obtained by two‐phase algorithms. Based on properties of the lexicographic breadth‐first search, we propose a new one‐phase algorithm. We present also a characterization for planar chordal graphs; using our proposed algorithm as an initial step, the implementation of the recognition algorithm becomes trivial.

Equilateral Drawing of 2-Connected Planar Chordal Graphs

Equilateral Drawing of 2-Connected Planar Chordal Graphs

Equilateral Drawing of 2-Connected Planar Chordal Graphs

Equilateral Drawing of 2-Connected Planar Chordal Graphs