arXiv subjects

Mario Valencia-Pabon

Publications and source records attributed to Mario Valencia-Pabon.

12 recordsLinked to original sources

arXiv2026

On the spectrum and expansion of graph associahedra

In this article, we contribute to the spectral analysis of graph associahedra by providing a lower bound for the second largest eigenvalue of $\mathcal{A}(G)$. Furthermore, using equitable partitions, we analyze the spectrum of the stellohedron $\mathcal{A}(K_{1,n})$. Specifically, we prove the existence of an eigenvalue in each interval $(n-i, n-i+1]$ for $1 \leq i \leq 5$, establish the presence of an eigenvalue with high multiplicity in $(n - \frac{3}{n} + \frac{2}{n^2-n}, n)$, and identify two additional small eigenvalues.

math.CO↗

arXiv2026

Token sliding independent set reconfiguration on graphs with few $P_4$'s

We consider the INDEPENDENT SET RECONFIGURATION problem under the Token Sliding rule. Let $I$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $I$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the vertices having tokens always form an independent set of $G$. The problem we deal is to decide if we can transform $I$ into $I'$ through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph "is reachable" from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, bipartite permutation graphs, block graphs, and cographs. We present a polynomial time algorithm for the problem on $P_4$-tidy graphs and $(q,q-4)$-graphs, both families of graphs generalizing cographs.

cs.DS↗

arXiv2025

Computing distances is FPT on graph associahedra and W[2]-hard on hypergraphic polytopes

An elimination tree of a connected graph $G$ is a rooted tree on the vertices of $G$ obtained by choosing a root $v$ and recursing on the connected components of $G-v$ to obtain the subtrees of $v$. The graph associahedron of $G$ is a polytope whose vertices correspond to elimination trees of $G$ and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where $G$ is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph $G$, is fixed-parameter tractable parameterized by the distance $k$. Prior to our work, only the case where $G$ is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of $k$. On the negative side, we show that it is unlikely that FPT algorithms exist on a natural generalization of graph associahedra, namely hypergraphic polytopes, by proving that computing distances on them is W[2]-hard parameterized by the distance. We also prove that, on hypergraphic polytopes, the distance cannot be approximated in polynomial time within a factor $c \cdot \log(|V|+|\mathcal{E}|)$ for some constant $c > 0$ unless P = NP, where $H=(V, \mathcal{E})$ is the input hypergraph. This result strengthens the hardness result of Cardinal and Steiner [Combin. Theory 2025], who proved that the problem cannot be approximated within a factor $(1 + \varepsilon)$ for some absolute constant $\varepsilon > 0$ unless P = NP. Finally, we rule out the existence of polynomial kernels parameterized by the number of vertices of the input hypergraph, a parameter for which the problem is easily seen to be FPT.

cs.DS↗

arXiv2024

Exact distance Kneser graphs

For any graph $G = (V,E)$ and positive integer $d$, the exact distance-$d$ graph $G_{=d}$ is the graph with vertex set $V$, where two vertices are adjacent if and only if the distance between them in $G$ is $d$. We study the exact distance-$d$ Kneser graphs. For these graphs, we characterize the adjacency of vertices in terms of the cardinality of the intersection between them. We present formulas describing the distance between any pair of vertices and we compute the diameter of these graphs.

math.CO↗

arXiv2023

The rotation distance of brooms

The associahedron $\mathcal{A}(G)$ of a graph $G$ has the property that its vertices can be thought of as the search trees on $G$ and its edges as the rotations between two search trees. If $G$ is a simple path, then $\mathcal{A}(G)$ is the usual associahedron and the search trees on $G$ are binary search trees. Computing distances in the graph of $\mathcal{A}(G)$, or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when $G$ is a complete split graph. In that case, $\mathcal{A}(G)$ interpolates between the stellohedron and the permutohedron, and all the search trees on $G$ are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of $G$ can be computed in quasi-quadratic time in the number of vertices of $G$.

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 diameter of Schrijver graphs

For $k \geq 1$ and $n \geq 2k$, the well known Kneser graph $\operatorname{KG}(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph $\operatorname{SG}(n,k)$ of $\operatorname{KG}(n,k)$ with the same chromatic number. In this paper, we compute the diameter of the graph $\operatorname{SG}(2k+r,k)$ with $r \geq 1$. We obtain an exact value of the diameter of $\operatorname{SG}(2k+r,k)$ when $r \in \{1,2\}$ or when $r \geq k-3$. For the remained cases, when $3 \leq r \leq k-4$, we obtain that the diameter of $\operatorname{SG}(2k+r,k)$ belongs to the integer interval $[4..k-r-1]$.

math.CO↗

arXiv2021

Diameter estimates for graph associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $Θ(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is $Θ(n\log n)$. Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.

math.CO↗

arXiv2016

Shifts of the Stable Kneser Graphs and Hom-Idempotence

A graph $G$ is said to be {\em hom-idempotent} if there is a homomorphism from $G^2$ to $G$, and {\em weakly hom-idempotent} if for some $n \geq 1$ there is a homomorphism from $G^{n+1}$ to $G^n$. Larose et al. [{\em Eur. J. Comb. 19:867-881, 1998}] proved that Kneser graphs $\operatorname{KG}(n,k)$ are not weakly hom-idempotent for $n \geq 2k+1$, $k\geq 2$. For $s \geq 2$, we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) of $s$-stable Kneser graphs $\operatorname{KG}(n,k)_{s-\operatorname{stab}}$ and we show that $2$-stable Kneser graphs are not weakly hom-idempotent, for $n \geq 2k+2$, $k \geq 2$. Moreover, for $s,k\geq 2$, we prove that $s$-stable Kneser graphs $\operatorname{KG}(ks+1,k)_{s-\operatorname{stab}}$ are circulant graphs and so hom-idempotent graphs. Finally, for $s \geq 3$, we show that $s$-stable Kneser graphs $\operatorname{KG}(2s+2,2)_{s-\operatorname{stab}}$ are cores, not $χ$-critical, not hom-idempotent and their chromatic number is equal to $s+2$.

math.CO↗

arXiv2014

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χ_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ(G),\ldots,χ_b(G), and b-monotonic if χ_b(H_1) \geq χ_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

cs.CC↗

arXiv2008

Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.

cs.DM↗

arXiv2006

A distributed approximation algorithm for the minimum degree minimum weight spanning trees

Fischer has shown how to compute a minimum weight spanning tree of degree at most $b Δ^* + \lceil \log\_b n\rceil$ in time $O(n^{4 + 1/\ln b})$ for any constant $b > 1$, where $Δ^*$ is the value of an optimal solution and $n$ is the number of nodes in the network. In this paper, we propose a distributed version of Fischer's algorithm that requires messages and time complexity $O(n^{2 + 1/\ln b})$, and O(n) space per node.

cs.DC↗