Proper Interval Graphs Research Articles

Unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties

Abstract The solution of Shareshian–Wachs conjecture by Brosnan–Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg–Buchstaber. We prove the palindromicity of LLT polynomials from topological viewpoint. We also show that modules of a symmetric group generated by faces of a permutohedron are related to a shifted unicellular LLT polynomial and observe the $e$-positivity of shifted unicellular LLT polynomials, which is established by Alexandersson–Sulzgruber in general, for path graphs and complete graphs through the cohomology of the twins.

Semi-proper orientations of dense graphs

An orientation D of a graph G is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation D of a graph G is a k-orientation if the in-degree of each vertex in D is at most k. An orientation D of G is proper if any two adjacent vertices have different in-degrees in D. The proper orientation number of a graph G, denoted by →χ (G), is the minimum k such that G has a proper k-orientation.A weighted orientation of a graph G is a pair (D, w), where D is an orientation of G and w is an arc-weighting A(D) → N \\ {0}. A semi-proper orientation of G is a weighted orientation (D, w) of G such that for every two adjacent vertices u and v in G, we have that S(d,w)(v) ≠ S(d,w)(u), where S(d,w)(v) is the sum of the weights of the arcs in (D, w) with head v. For a positive integer k, a semi-proper k-orientation (D, w) of a graph G is a semi-proper orientation of G such that maxvϵV(G) S(d,w)(v) ≤ k. The semi-proper orientation number of a graph G, denoted by →χs(G), is the least k such that G has a semi-proper k-orientation.In this work, we first prove that →χs(G) ϵ {ω(G) - 1, ω(G)} for every split graph G, and that, given a split graph G, deciding whether →χs(G) = ω(G) - 1 is an NP-complete problem. We also show that, for every k, there exists a (chordal) graph G and a split subgraph H of G such that →χ(G) ≤ k and →χ(H) = 2k - 2. In the sequel, we show that, for every n ≥ p(p + 1), →χs(Ppn) = [3/2 p], where Ppn is the pth power of the path on n vertices. We investigate further unit interval graphs with no big clique: we show that →χ(G) ≤ 3 for any unit interval graph G with ω(G) = 3, and present a complete characterization of unit interval graphs with →χ(G)= ω(G) = 3. Then, we show that deciding whether →χs(G) = ω(G) can be solved in polynomial time in the class of co-bipartite graphs. Finally, we prove that computing →χs(G) is FPT when parameterized by the minimum size of a vertex cover in G or by the treewidth of G. We also prove that not only computing →χs(G) but also →χ(G), admits a polynomial kernel when parameterized by the neighbourhood diversity plus the value of the solution. These results imply kernels of size 40(k2) and 0(2kk2), in chordal graphs and split graphs, respectively, for the problem of deciding whether →χs(G) ≤ k parameterized by k. We also present exponential kernels for computing both →χ(G) and →χs(G) parameterized by the value of the solution when G is a cograph. On the other hand, we show that computing →χs(G) does not admit a polynomial kernel parameterized by the value of the solution when G is a chordal graph, unless NP ⊆ coNP/poly.

Acyclic matching in some subclasses of graphs

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. Given a matching M in G, a vertex v∈V is called M-saturated if there exists an edge e∈M incident with v. A matching M of a graph G is called an acyclic matching if, G[V(M)], the subgraph of G induced by the M-saturated vertices of G is an acyclic graph. Given a graph G, the Acyclic Matching problem asks to find an acyclic matching of maximum cardinality in G. The Decide-Acyclic Matching problem takes a graph G and an integer k and asks whether G has an acyclic matching of cardinality at least k. The Decide-Acyclic Matching problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen this result by showing that the Decide-Acyclic Matching problem remains NP-complete for comb-convex bipartite graphs, star-convex bipartite graphs, and dually chordal graphs. On the positive side, we show that the Acyclic Matching problem is linear time solvable for split graphs, block graphs, and proper interval graphs. We show that the Acyclic Matching problem is hard to approximate within a factor of n1−ϵ for any ϵ>0 unless P=NP. Also, we show that the Acyclic Matching problem is APX-complete for (2k+1)-regular graphs for every fixed integer k≥3.

Precedence thinness in graphs

Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have been introduced generalizing interval and proper interval graphs, respectively. The complexity of the recognition of each of these classes is still open, even for fixed $k \geq 2$. In this work, we introduce a subclass of $k$-thin graphs (resp. proper $k$-thin graphs), called precedence $k$-thin graphs (resp. precedence proper $k$-thin graphs). Concerning partitioned precedence $k$-thin graphs, we present a polynomial time recognition algorithm based on $PQ$-trees. With respect to partitioned precedence proper $k$-thin graphs, we prove that the related recognition problem is \NP-complete for an arbitrary $k$ and polynomial-time solvable when $k$ is fixed. Moreover, we present a characterization for these classes based on threshold graphs.

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.

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.

Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Pósa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and (tight and weak) Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1)Every unit-interval strongly-connected d-dimensional simplicial complex is traceable.(This extends the well-known result “unit-interval connected graphs are traceable”.)(2)Every unit-interval d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian.(This extends the fact that “unit-interval 2-connected graphs are Hamiltonian”.)(3)Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Gröbner basis for a diagonal term order which is compatible with the traceability of the complex.(This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa–Vandebogert.)(4)Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution.(This extends the result by Kiani and Saeedi-Madani that “only the complete graph has a binomial edge ideal with linear resolution”.)(5)The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.

Positivity of Chromatic Symmetric Functions Associated with Hessenberg Functions of Bounce Number 3

We give a proof of the Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials GP(x;q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q+1, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

The co-secure domination in proper interval graphs

A set S⊂V is a co-secure dominating set of a graph G=(V,E) if S is a dominating set, and for each u∈S there exists a vertex v∈V∖S such that uv∈E and (S∖{u})∪{v} is a dominating set. Note that |V|>1. The minimum cardinality of a co-secure dominating set in G is the co-secure domination number of G. In this paper, we propose a linear-time algorithm for finding the co-secure domination number of proper interval graphs.

The hull number in the convexity of induced paths of order 3

A set S of vertices of a graph G is P3⁎-convex if there is no vertex outside S having two non-adjacent neighbors in S. The P3⁎-convex hull of S is the minimum P3⁎-convex set containing S. If the P3⁎-convex hull of S is V(G), then S is a P3⁎-hull set. The minimum size of a P3⁎-hull set is the P3⁎-hull number of G. In this paper, we show that the problem of deciding whether the P3⁎-hull number of a chordal graph is at most k is NP-complete and present a linear-time algorithm to determine this parameter and provide a minimum P3⁎-hull set for unit interval graphs.

Length-bounded cuts: Proper interval graphs and structural parameters

We study the Length-Bounded Cut problem for special graph classes and from a parameterized complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set F of at most β edges such that each s-t-path of length at most λ in G contains some edge in F. Bazgan et al. [20] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph is a proper interval graph. We confirm this conjecture by providing a dynamic-programming-based polynomial-time algorithm. Moreover, we strengthen the W[1]-hardness result of Dvořák and Knop [15] for Length-Bounded Cut parameterized by pathwidth by showing W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.

Efficient algorithms for independent Roman domination on some classes of graphs

Let $G=(V,E)$ be a given graph of order $n $. A function $f : V to {0,1, 2}$ is an independent Roman dominating function (IRDF) on $G$ if for every vertex $vin V$ with $f(v)=0$ there is a vertex $u$ adjacent to $v$ with $f(u)=2$ and ${vin V:f(v)> 0}$ is an independent set. The weight of an IRDF $f$ on $G $ is the value $f(V)=sum_{vin V}f(v)$. The minimum weight of an IRDF among all IRDFs on $G$ is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of $G$ in $O(|V|)$ time when $G=(V,E)$ is a tree, unicyclic graph or proper interval graph.

On the maximum cardinality cut problem in proper interval graphs and related graph classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this work we consider the parameterized complexity of this problem. We show that the maximum cardinality cut problem in proper/unit interval graphs is FPT when parameterized by the maximum number of non-empty bubbles in a column of its bubble model. We then generalize this result to a more general graph class by defining new parameters related to the well-known clique-width parameter.Specifically, we define an (α,β,δ)-clique-width decomposition of a graph as a clique-width decomposition in which at each step the following invariant is preserved: after discarding at most δ labels, a) every label consists of at most β sets of twin vertices, and b) all the labels together induce a graph with independence number at most α. We show that for every two constants α,δ>0 the problem is FPT when parameterized by β plus the smallest width of an (α,β,δ)-clique-width decomposition.

{mathcal {U}}-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. https://doi.org/10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

A Theorem on Indifference Graphs

An indifference graph P has as vertices a set of points on the real line, with an edge between every two points at most 1 apart. For every point x in P, we set N(x) to be the set of points that are adjacent to x. Fix an indifference graph P, and a positive integer k. Suppose that for every x in P, k divides |N(x)|. We show that the number of points in P is also divisible by k.

Weighted total acquisition

In the Weighted Total Acquisition problem (WTA) on a weighted graph G (only positive non-zero weights), a vertex v can acquire the total weight of a neighbour u if and only if the current weight of v is at least that of u. The new weight of u is then zero, and the new weight of v is then the sum of the weights at u and v just before the acquisition. Over all possible acquisition sequences in G with weight function w, the minimum number of vertices with a non-zero weight at the end is denoted by at(G,w). Given a graph G, a weighting w, and an integer k≥1, the WTA problem asks whether at(G,w)≤k. The Unary (Binary resp.) WTA problem corresponds to the WTA problem when the weights are encoded in unary (binary resp.).We prove that Unary WTA is polynomial-time solvable in graphs of bounded treewidth and degree. When only the treewidth is bounded, this algorithm is quasi-polynomial, i.e., it runs in time WO(logW), where W is the sum of the weights of the vertices. Moreover, we show that Unary WTA is FPT in trees when parameterized by the maximum degree. On the negative side, we show that Unary WTA is NP-complete in trivially perfect graphs and split graphs, even when k=1 in the latter.We prove that the Binary WTA problem is NP-complete in trees of bounded degree, trees of bounded depth, and wheels, but that it is in XP for trees and wheels when parameterized by the solution size. Moreover, we show that Binary WTA is NP-complete in K3,n, planar graphs of pathwidth 2, and unit interval graphs even when k=1, and in trivially perfect graphs when k≥2 (but polynomial-time solvable when k=1).

Total 2-domination of proper interval graphs

A set of vertices W of a graph G is a total k-dominating set when every vertex of G has at least k neighbors in W. In a recent article, Chiarelli et al. (2019) prove that a total k-dominating set can be computed in O(n3k) time when G is a proper interval graph with n vertices and m edges. In this note we reduce the time complexity to O(m) for k=2.

On the complexity of minimum maximal uniquely restricted matching

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. A matching M in a graph G is called a uniquely restricted matching if, G[V(M)], the subgraph of G induced by the set of M-saturated vertices of G contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching in G of minimum cardinality and Decide-Min-Max-UR Matching problem, the decision version of this problem takes a graph G and an integer k and asks whether G admits a maximal uniquely restricted matching of cardinality at most k. It is known that the Decide-Min-Max-UR Matching problem is NP-complete. In this paper, we strengthen this result by proving that the Decide-Min-Max-UR Matching problem remains NP-complete for chordal bipartite graphs, star-convex bipartite graphs, chordal graphs, and doubly chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite distance-hereditary graphs and linear time solvable for bipartite permutation graphs, proper interval graphs, and threshold graphs. Finally, we prove that the Min-Max-UR Matching problem is APX-complete for graphs with maximum degree 4.

On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Motivated by Stanley and Stembridge's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which are two important families of $(claw, 2K_2)$-free graphs. Hence we affirmatively solve one problem proposed by Hamel, Hoáng and Tuero, and another problem considered by Foley, Hoáng and Merkel.