Edge Domination Problem Research Articles

Extended formulations for perfect domination problems and their algorithmic implications

Given an undirected graph G=(V,E), a subset D⊆V is called a vertex dominating set (VDS) if every vertex of V either belongs to D or is adjacent to a vertex of D. Additionally, a VDS D is called perfect if every vertex of V∖D is adjacent to a single vertex of D. Finally, the Perfect Vertex Domination Problem (PVDP) asks for a perfect VDS D with the smallest cardinality possible. Domination extends very naturally to the edges of G=(V,E) and the Perfect Edge Domination Problem (PEDP) asks for a perfect edge dominating set with as few edges as possible. We propose new formulations for PVDP and PEDP. They rely on structural properties of perfect dominating sets and are computationally compared with their counterparts from the literature. For the new PEDP formulation, in particular, running times for standard state-of-the-art mixed integer programming codes are shown to frequently lead to speed-ups of orders of magnitude, over their corresponding performances for the remaining PEDP formulations.

The complexity of total edge domination and some related results on trees

For a graph \(G = (V, E)\) with vertex set V and edge set E, a subset F of E is called an edge dominating set (resp. a total edge dominating set) if every edge in \(E\backslash F\) (resp. in E) is adjacent to at least one edge in F, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number) of G, denoted by \(\gamma ^{\prime }(G)\) (resp. \(\gamma _t^{\prime }(G)\)). In the present paper, we first prove that the total edge domination problem is NP-complete for bipartite graphs with maximum degree 3. Then, for a graph G, we give the inequality \(\gamma ^{\prime }(G)\leqslant \gamma ^{\prime }_{t}(G)\leqslant 2\gamma ^{\prime }(G)\) and characterize the trees T which obtain the upper or lower bounds in the inequality.

Finding dominating induced matchings in [formula omitted]-free graphs in polynomial time

Let G=(V,E) be a finite undirected graph. An edge set E′⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E′. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem; it is the Efficient Domination problem for line graphs.The DIM problem is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for P7-free graphs, and in polynomial time for S1,2,4-free graphs as well as for S2,2,2-free graphs and for S2,2,3-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for S1,1,5-free graphs.

Finding dominating induced matchings in [formula omitted]-free graphs in polynomial time

Let G=(V,E) be a finite undirected graph. An edge set E′⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E′. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem; it is the Efficient Domination problem for line graphs. The DIM problem is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 and is solvable in linear time for P7-free graphs, and in polynomial time for S1,2,4-free graphs as well as for S2,2,2-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for S2,2,3-free graphs.

Dominating induced matchings in [formula omitted]-free graphs

Let G=(V,E) be a finite undirected graph without loops and multiple edges. A subset M⊆E of edges is a dominating induced matching (d.i.m. ) in G if every edge in E is intersected by exactly one edge of M. In particular, this means that M is an induced matching, and every edge not in M shares exactly one vertex with an edge in M. Clearly, not every graph has a d.i.m.The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem; it is the Efficient Domination problem for line graphs.The DIM problem is NP-complete in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomial time for claw-free (i.e., S1,1,1-free) graphs, for S1,2,3-free graphs, for S2,2,2-free graphs as well as for S2,2,3-free graphs, in linear time for P7-free graphs, and in polynomial time for P8-free graphs (Pk is a special case of Si,j,ℓ). In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial time for Si,j,k-free graphs for every fixed i,j,k.In this paper, combining two distinct approaches, we solve it in polynomial time for S1,2,4-free graphs which generalizes the S1,2,3-free as well as the P7-free case.

Modelling and solving the perfect edge domination problem

A formulation is proposed for the perfect edge domination problem and some exact algorithms based on it are designed and tested. So far, perfect edge domination has been investigated mostly in computational complexity terms. Indeed, we could find no previous explicit mathematical formulation or exact algorithm for the problem. Furthermore, testing our algorithms also represented a challenge. Standard randomly generated graphs tend to contain a single perfect edge dominating solution, i.e., the trivial one, containing all edges in the graph. Accordingly, some quite elaborated procedures had to be devised to have access to more challenging instances. A total of 736 graphs were thus generated, all of them containing feasible solutions other than the trivial ones. Every graph giving rise to a weighted and a non weighted instance, all instances solved to proven optimality by two of the algorithms tested.

Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset \(E'\) of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of \(E'\). We say that \(E'\) is a perfect edge dominating set of G, if every edge not in \(E'\) is dominated by exactly one edge of \(E'\). The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most \(d \ge 3\) and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a \(P_5\)-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a \(P_5\).

Finding Dominating Induced Matchings in $$P_8$$ P 8 -Free Graphs in Polynomial Time

Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.

Dominating Induced Matchings for P 7-Free Graphs in Linear Time

Let G be a finite undirected graph with edge set E. An edge set E′⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge e∈E∖E′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is ${\mathbb{NP}}$ -complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for P k -free graphs for any k≥5; P k denotes a chordless path with k vertices and k−1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7-free graphs in a robust way.

The rook problem on saw-toothed chessboards

A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases from left down to right without any hole inside it. A rook at square ( i , j ) can dominate the squares in row i and in column j . The rook problem of an STC is to determine the minimum number of rooks that can dominate all squares of the STC. In this paper, we model an STC by two particular graphs: a rook graph and a board graph. We show that for an STC, the rook graph is the line graph of the board graph, and the board graph is a bipartite permutation graph. Thus, the rook problem on STCs can be solved by any algorithm for solving the edge domination problem on bipartite permutation graphs.

Perfect edge domination and efficient edge domination in graphs

Let G=( V, E) be a finite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E⧹ D is dominated by exactly one edge in D and an efficient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (efficient) edge domination problem is to find a perfect (efficient) edge dominating set of minimum size in G. Suppose that each edge e is associated with a real number w( e) as its weight. Then, the weighted perfect (efficient) edge domination problem is to calculate a perfect (efficient) edge dominating set D such that the weight w( D) of D is minimum, where w( D)=∑ e∈ D w( e). In this paper, we show that the perfect (efficient) edge domination problem is NP-complete on bipartite (planar bipartite) graphs. Moreover, we present linear-time algorithms to solve the weighted perfect (efficient) edge domination problem on generalized series–parallel graphs and chordal graphs.

Solving the weighted efficient edge domination problem on bipartite permutation graphs

Given a simple graph G = ( V, E), an edge ( u, v) ϵ E is said to dominate itself and any edge ( u, x) or ( v, x), where xϵ V. A subset D⊂- E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge eϵ E is associated with a real number w( e), called the weight of e. The weighted efficient edge domination problem is to calculate an efficient edge dominating set D of G such that the weight w( D) of D is minimum, where w(D) = ∑{w(e)¦eϵD}. In this paper, we show that the problem of determining whether G has an efficient edge dominating set is NP-complete when G is restricted to a bipartite graph. Consequently, the decision problem of efficient (vertex) domination remains NP-complete for the line graphs of bipartite graphs. Moreover, we present a linear time algorithm to solve the weighted efficient edge domination problem on bipartite permutation graphs, which form a subclass of bipartite graphs, using the technique of dynamic programming.

Quasi-threshold graphs

Quasi-threshold graphs are defined recursively by the following rules: 1. (1) K 1 is a quasi-threshold graph, 2. (2) adding a new vertex adjacent to all vertices of a quasi-threshold graph results in a quasi-threshold graph, 3. (3) the disjoint union of two quasi-threshold graphs is a quasi-threshold graph. This paper gives some new equivalent definitions of a quasi-threshold graph. From them, linear time recognition algorithms follow. We also give linear time algorithms for the edge domination problem and the bandwidth problem in this class of graphs.

Minimum Edge Dominating Sets

Let $G = ( V,E )$ be a finite undirected graph with n vertices and m edges. A minimum edge dominating set of G is a set of edges D, of smallest cardinality $\gamma ' ( G )$, such that each edge of $E - D$ is adjacent to some edge of D. Let $S( G )$ be the subdivision graph of G and let $T( G )$ be the total graph of G. Let $\alpha ( G )$ be the stability number of G (cardinality of a largest stable set) and let $\alpha _2 ( G )$ be the 2-stability number of G (cardinality of a largest set of vertices in G, no two of which are joined by a path of length 2 or less). The following results are obtained. For any $G,\gamma' ( S ( G ) ) + \alpha _2 ( G ) = n$ and $2\gamma ' ( T ( G ) ) + \alpha ( T ( G ) ) = n + m$ or $n + m + 1$. Also, for any depth-first search tree S of $G,\gamma ' ( S )/2\leqq \gamma ' ( G )\leqq 2\gamma ' (S)$, and these bounds are tight.The edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cub...