Class Of Proper Interval Graphs Research Articles

Exactly Hittable Interval Graphs

Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.

On central max-point-tolerance graphs

Max-point-tolerance graphs (MPTG) were studied by Catanzaro et al. in 2017 and the same class of graphs were introduced in the name of p-BOX(1) graphs by Soto and Caro in 2015. This class has a wide application in genome studies as well as in telecommunication networks. In our article, we consider central max-point-tolerance graphs (central MPTG) by taking the points of MPTG as center points of their corresponding intervals. In the course of study on this class of graphs, we show that the class of central MPTG is same as the class of unit max-tolerance graphs. We also prove that the class of unit central MPTG is same as that of proper central MPTG and both of them are equivalent to the class of proper interval 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.

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.

A decoupled local memory allocator

Compilers use software-controlled local memories to provide fast, predictable, and power-efficient access to critical data. We show that the local memory allocation for straight-line, or linearized programs is equivalent to a weighted interval-graph coloring problem. This problem is new when allowing a color interval to “wrap around,” and we call it the submarine-building problem. This graph-theoretical decision problem differs slightly from the classical ship-building problem, and exhibits very interesting and unusual complexity properties. We demonstrate that the submarine-building problem is NP-complete, while it is solvable in linear time for not-so-proper interval graphs, an extension of the the class of proper interval graphs. We propose a clustering heuristic to approximate any interval graph into a not-so-proper interval graph, decoupling spill code generation from local memory assignment. We apply this heuristic to a large number of randomly generated interval graphs reproducing the statistical features of standard local memory allocation benchmarks, comparing with state-of-the-art heuristics.

Approximating the path-distance-width for AT-free graphs and graphs in related classes

We consider the problem of determining the path-distance-width for AT-free graphs and graphs in related classes such as k-cocomparability graphs, proper interval graphs, cobipartite graphs, and cochain graphs. We first show that the problem is NP-hard even for cobipartite graphs, and thus for AT-free graphs. Next we present simple approximation algorithms with constant approximation ratios for AT-free graphs and graphs in the related graph classes mentioned above. For instance, our algorithm for AT-free graphs has approximation factor 3 and runs in linear time. We also show that the problem is solvable in polynomial time for cochain graphs, which form a subclass of the class of proper interval graphs.

Subgraph isomorphism in graph classes

We investigate the computational complexity of the following restricted variant of Subgraph Isomorphism: given a pair of connected graphs G=(VG,EG) and H=(VH,EH), determine if H is isomorphic to a spanning subgraph of G. The problem is NP-complete in general, and thus we consider cases where G and H belong to the same graph class such as the class of proper interval graphs, of trivially perfect graphs, and of bipartite permutation graphs. For these graph classes, several restricted versions of Subgraph Isomorphism such as Hamiltonian Path, Clique, Bandwidth, and Graph Isomorphism can be solved in polynomial time, while these problems are hard in general.

Short Models for Unit Interval Graphs

Abstract We present one more proof of the fact that the class of proper interval graphs is precisely the class of unit interval graphs. The proof leads to a new and efficient O ( n ) time and space algorithm that transforms a proper interval model of the graph into a unit model, where all the extremes are integers in the range 0 to O ( n 2 ) , solving a problem posed by Gardi (Discrete Math., 307 (22), 2906–2908, 2007).

A polynomial solution to the [formula omitted]-fixed-endpoint path cover problem on proper interval graphs

We study a variant of the path cover problem, namely, the k -fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V ( G ) , a k -fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P . The kPC problem is to find a k -fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k = 0 ), the stated problem coincides with the classical path cover problem. The kPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete. Note that the complexity status for both 1HP and 2HP problems on interval graphs remains an open question (Damaschke (1993) [9]). In this paper, we show that the kPC problem can be solved in linear time on the class of proper interval graphs, that is, in O ( n + m ) time on a proper interval graph on n vertices and m edges. The proposed algorithm is simple, requires linear space, and also enables us to solve the 1HP and 2HP problems on proper interval graphs within the same time and space complexity.

A polynomial algorithm for the k-cluster problem on the interval graphs

This paper deals with the problem of finding, for a given graph and a given natural number k, a subgraph of k nodes with a maximum number of edges. This problem is known as the k-cluster problem and it is NP-hard on general graphs as well as on chordal graphs. In this paper, it is shown that the k-cluster problem is solvable in polynomial time on interval graphs. In particular, we present two polynomial time algorithms for the class of proper interval graphs and the class of general interval graphs, respectively. Both algorithms are based on a novel matrix representation for interval graphs. In contrast to representations used in most of the previous work, this matrix representation does not make use of the maximal cliques in the investigated graph.

Minimum proper interval graphs

A graph G is a proper interval graph if there exists a mapping r from V( G) to the class of closed intervals of the real line with the properties that for distinct vertices v and w we have r(v) ⋔ r(w) ≠ Ø if and only if v and w are adjacent and neither of the intervals r( v), r( w) contain the other. We prove that for every proper interval graph G, | V( G)| ⩾ 2 c( G) - c( K( G)), where c( G) is the number of cliques of G and K( G) is the clique graph of G. If the equality is verified we call G a minimum proper interval graph. The main result is that the restriction to the class of minimum proper interval graphs of clique mapping G → K( G) is a bijection (up to isomorphism) onto the class of proper interval graphs. We find the greatest clique-closed class Σ ( K( Σ) = Σ) contained in the union of the class of connected minimum proper interval graphs and the class of complete graphs. We enumerate the minimum proper interval graphs with n vertices.

Interval graphs and related topics

Interval graphs have engaged the interest of researchers for over twenty-five years. The scope of current research in this area now extends to the mathematical and algorithmic properties of interval graphs, their generalizations, and graph parameters motivated by them. One main reason for this increasing interest is that many real-world applications involve solving problems on graphs which are either interval graphs themselves or are related to interval graphs in a natural way. The problem of characterizing interval graphs was first posed independently by Hajos [45] in combinatorics and by Benzer [17] in genetics. By definition, an interual graph is the intersection graph of a family of intervals on the real line, i.e., to every vertex of the graph there corresponds an interval and two vertices are connected by an edge of the graph if and only if their corresponding intervals have nonempty intersection. However, this suggests a more general paradigm for studying various classes of graphs which can be described as follows. Let9=[&,..., S,] be a family of nonempty subsets of a set S. The subsets are not necessarily distinct. We will call S the host and the subsets Si the objects. In addition, there may or may not be certain constraints placed on the objects such as not allowing one subset to properly contain another or requiring that each subset satisfies a special property. The intersection graph of 9 has vertices Ul,...? II,, with Ui and vi joined by an edge if and only if Si n Sj # 0. We call the pair X = (S, 9) an intersection representation hypergruph for G or more simply a representation. When 9 is a family of intervals on a line, G is an interval graph and X is an interval hypergraph. If we add the constraint that no interval may properly contain another, then we obtain the class of proper interval graphs. When 9 is allowed to be an arbitrary family of sets, the class obtained as intersection graphs is all undirected graphs. On one hand, research has been directed towards intersection graphs of families having some specific topological or other structure. These include circular-arcs, paths in trees, chords of circles, cliques of graphs, and others [35]. On the other hand, certain well-known classes of graphs have subsequently been characterized in terms of intersection graphs. For example, triangulated graphs (every cycle of length 24 has a chord) are the intersection graphs of subtrees of a tree [21, 33,