Searcharxiv⌕ Search

arXiv subjects

Luciano N. Grippo

Publications and source records attributed to Luciano N. Grippo.

At least 19 recordsLinked to original sources
arXiv2026

Counting perfect edge dominating sets: extremal results and linear-time algorithms

An edge of a graph \emph{dominates} itself and each edge adjacent to it. A \emph{perfect edge dominating set} is a subset of edges such that each edge outside the subset is dominated by exactly one edge of the subset. In this article, we characterize the extremal graphs on $n$ vertices in the classes of trees, forests, and chordal graphs with respect to the number of perfect edge dominating sets. Moreover, we derive linear-time algorithms for counting perfect edge dominating sets and for counting dominating induced matchings in generalized series-parallel graphs and chordal graphs.

math.CO↗
arXiv2026

A Sharp Forbidden Interval for the Nontrivial Adjacency Eigenvalues of Trivially Perfect Graphs

We prove a sharp forbidden interval for the nontrivial adjacency eigenvalues of trivially perfect graphs. More precisely, we show that if $G$ is a trivially perfect graph, then $\operatorname{Spec}(G)\cap [\sqrt{8}-4,0]\subseteq \{-1,0\}$. Moreover, we prove that the interval is best possible at both endpoints: there are connected trivially perfect graphs with eigenvalues arbitrarily close to $\sqrt{8}-4$ from below, and connected trivially perfect graphs with positive eigenvalues converging to $0$.

math.SP↗
arXiv2026

Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets

An edge of a graph dominates itself along with any edge that shares an endpoint with it. An efficient edge dominating set (also called a dominating induced matching, DIM) is a subset of edges such that each edge of the graph is dominated by exactly one edge in the subset. A perfect edge dominating set is a subset of edges in which every edge outside the subset is dominated by exactly one edge within it. In this article, we establish the NP-completeness of deciding whether a graph that does not admit any efficient edge dominating set has at least two perfect edge dominating sets. We also present a cubic time algorithm designed to identify a perfect dominating set of minimal cardinality for $P_6$-free graphs. Moreover, we show how this algorithm can be adapted to handle the weighted version of the problem and to count all perfect edge dominating sets as well as DIMs in a given graph, while preserving the same time complexity.

math.CO↗
arXiv2026

Counting $P_3$-convex sets in graphs

We study the $P_3$-convexity, the path convexity generated by all three-vertex paths, and focus on the problem of counting the $P_3$-convex vertex sets of a graph $G$, denoted by $\noc(G)$. First, we settle the associated extremal question: we characterize the $n$-vertex graphs maximizing $\noc(G)$ among all graphs and determine the connected extremal graphs. Next, we investigate computational complexity and show that counting $P_3$-convex sets is $\#\mathsf{P}$-complete already on split graphs, even under additional structural restrictions. On the positive side, we identify two tractable subclasses, namely trees and threshold graphs, and obtain linear-time algorithms for both. Finally, we design nontrivial exact exponential-time algorithms for general graphs, combining structural decomposition, propagation rules capturing forced consequences of $P_3$-convexity, and fast counting of independent sets in auxiliary graphs. The resulting strategy becomes particularly effective on graph classes where large independent sets are guaranteed and can be found efficiently.

math.CO↗
arXiv2025

Formulas and Upper Bounds for the Carath{é}odory Number of Hamming Graphs

Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest $P_3$-convex set of $G$ that contains $S$. Carathéodory number of a graph $G$, denoted by $c(G)$, is the smallest integer $c$ such that for every subset $S \subseteq V(G)$ and every vertex $p$ in the $P_3$-hull of $S$, there exists a subset $F \subseteq S$ with $|F| \leq c$ such that $p$ belongs to the $P_3$-hull of $F$. In this article, we present upper bounds and formulas for the $P_3$-Carathéodory number in Hamming graphs, which are defined as the Cartesian product of $n$ complete graphs.

math.CO↗
arXiv2024

On the spectral radius of unbalanced signed bipartite graphs

A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$, $-1$, or $1$, depending on whether $ij$ represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph $\dot{G}$ is less or equal to the index of the adjacency matrix of its underlying graph $G$, i.e., $λ_1(\dot{G}) \le λ_1(G)$. Indeed, if $\dot{G}$ is balanced, then $λ_1(\dot{G})=λ_1(G)$. This inequality becomes strict when $\dot{G}$ is an unbalanced signed graph. Recently, Brunetti and Stanić found the whole list of unbalanced signed graphs on $n$ vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on $n$ vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on $n$ vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.

math.CO↗
arXiv2024

Singularly cospectral circulant graphs

Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse may be false. In this paper, we present sufficient conditions for two circulant graphs, with an even number of vertices, to be noncospectral singularly cospectral graphs. In this analysis, we study when a pair of these graphs have the same or distinct inertia. In addition, we show that two singularly cospectral circulant graphs with an odd prime number of vertices are isomorphic.

math.CO↗
arXiv2022

On restricted completions of chordal and trivially perfect graphs

Let $G$ be a graph having a vertex $v$ such that $H = G - v$ is a trivially perfect graph. We give a polynomial-time algorithm for the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a trivially perfect graph. This is a slight variation of the well-studied {\sc Edge Completion}, also known as {\sc Minimum Fill-In}, problem. We also show that if $H$ is a chordal graph, then the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a chordal graph is \NP-complete.

math.CO↗
arXiv2022

On the rank of the distance matrix of graphs

Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each integer $k\ge 2$, there is a finite amount of graphs whose distance matrices have rank $k$. We exhibit the list of graphs with distance matrices of rank $2$ and $3$. Besides, we study the rank of the distance matrices of graphs belonging to a family of graphs with their diameters at most two, the trivially perfect graphs. We show that for each $η\ge 1$ there exists a trivially perfect graph with nullity $η$. We also show that for threshold graphs, which are a subfamily of the family of trivially perfect graphs, the nullity is bounded by one.

math.CO↗
arXiv2022

On the generalized Helly property of hypergraphs, cliques, and bicliques

A family of sets is $(p,q)$-intersecting if every nonempty subfamily of $p$ or fewer sets has at least $q$ elements in its total intersection. A family of sets has the $(p,q)$-Helly property if every nonempty $(p,q)$-intersecting subfamily has total intersection of cardinality at least $q$. The $(2,1)$-Helly property is the usual Helly property. A hypergraph is $(p,q)$-Helly if its edge family has the $(p,q)$-Helly property and hereditary $(p,q)$-Helly if each of its subhypergraphs has the $(p,q)$-Helly property. A graph is $(p,q)$-clique-Helly if the family of its maximal cliques has the $(p,q)$-the Helly property and hereditary $(p,q)$-clique-Helly if each of its induced subgraphs is $(p,q)$-clique-Helly. The classes of $(p,q)$-biclique-Helly and hereditary $(p,q)$-biclique-Helly graphs are defined analogously. We prove several characterizations of hereditary $(p,q)$-Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. We give an improved time bound for the recognition of $(p,q)$-Helly hypergraphs for each fixed $q$ and show that the recognition of hereditary $(p,q)$-Helly hypergraphs can be solved in polynomial time if $p$ and $q$ are fixed but co-NP-complete if $p$ is part of the input. In addition, we generalize to $(p,q)$-clique-Helly graphs the characterization of $p$-clique-Helly graphs in terms of expansions and give different characterizations of hereditary $(p,q)$-clique-Helly graphs, including one by forbidden induced subgraphs. We give an improvement on the time bound for the recognition of $(p,q)$-clique-Helly graphs and prove that the recognition problem of hereditary $(p,q)$-clique-Helly graphs is polynomial-time solvable for $p$ and $q$ fixed but NP-hard if $p$ or $q$ is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (hereditary) $(p,q)$-biclique-Helly graphs.

math.CO↗
arXiv2020

Finding singularly cospectral graphs

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree.

math.CO↗
arXiv2020

On the spectral radius of block graphs having all their blocks of the same size

Let $\mathcal{B}(n,q)$ be the class of block graphs on $n$ vertices having all their blocks of the same size. We prove that if $G\in \mathcal{B}(n,q)$ has at most three pairwise adjacent cut vertices then the minimum spectral radius $ρ(G)$ is attained at a unique graph. In addition, we present a lower bound for $ρ(G)$ when $G\in \mathcal{B}(n,q)$.

math.SP↗
arXiv2020

Formulas in connection with parameters related to convexity of paths on three vertices: caterpillars and unit interval graphs

We present formulas to compute the P3-geodetic number, the P3-hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit interval graph and a parameter involving the diameter of a unit interval graph related to it. Finally, we present a hereditary graph class, defined by forbidden induced subgraphs, such that its percolation time is equal to one.

math.CO↗
arXiv2019

The determinant of the distance matrix of graphs with at most two cycles

Let $G$ be a connected graph on $n$ vertices and $D(G)$ its distance matrix. The formula for computing the determinant of this matrix in terms of the number of vertices is known when the graph is either a tree or {a} unicyclic graph. In this work we generalize these results, obtaining the determinant of the distance matrix for {all graphs} in a {class, including trees, unicyclic and bicyclic graphs. This class actually includes graphs with many cycles, provided that each block of the graph is at most bicyclic.}

math.CO↗
arXiv2019

On the spectral radius of graphs: nonregular distance-hereditary graphs with given edge-connectivity, graphs with tree-width $k$ and block graphs with prescribed independence number $α$

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $Δ(G)$ the maximum degree of a graph $G$ and let $ρ(G)$ be the spectral radius of $G$. In this article we present a lower bound for $Δ(G)-ρ(G)$ in terms of the edge connectivity of $G$, where $G$ is a nonregular distance-hereditary graph. We also prove that $ρ(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded tree-width or is in the class of block graphs with prescribed independence number.

math.CO↗
arXiv2019

On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphs

A $(0,1)$-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a $(0, 1)$-matrix is nested if it has the consecutive ones property for the rows (C1P) and every two rows are either disjoint or nested. We say a $(0, 1)$-matrix is 2-nested if it has the C1P and admits a partition of its rows into two sets such that the submatrix induced by each of these sets is nested. We say a split graph $G$ with split partition $(K, S)$ is nested (resp.\ 2-nested) if the matrix $A(S, K)$ which indicates the adjacency between vertices in $S$ and $K$ is nested (resp.\ 2-nested). In this work, we characterize nested and 2-nested matrices by minimal forbidden submatrices. This characterization leads to a minimal forbidden induced subgraph characterization for these classes of graphs, which are a superclass of threshold graphs and a subclass of split and circle graphs.

cs.DM↗
arXiv2017

Partial characterization of graphs having a single large Laplacian eigenvalue

The parameter $σ(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $σ(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $σ(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to some classes of graphs.

math.CO↗