arXiv subjects

Sahab Hajebi

Publications and source records attributed to Sahab Hajebi.

4 recordsLinked to original sources

arXiv2026

Asymmetric induced saturation

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs -- where both adding and deleting any edge creates an induced copy of $H$ -- appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.

math.CO↗

arXiv2025

Parameterized Complexity of the Star Decomposition Problem

A star of length $ \ell $ is defined as the complete bipartite graph $ K_{1,\ell } $. In this paper we deal with the problem of edge decomposition of graphs into stars of varying lengths. Given a graph $ G $ and a list of integers $S=(s_1,\ldots, s_t) $, an $S$-star decomposition of $ G $ is an edge decomposition of $ G $ into graphs $G_1 ,G_2 ,\ldots,G_t $ such that $G_i$ is isomorphic to an star of length $s_i$, for each $i \in\{1,2,\ldots,t\}$. Given a graph $G$ and a list of integers $S$, the \sdp problem asks if $G$ admits an $ S $-star decomposition. The problem is known to be NP-complete even when all stars are of length three. In this paper, we investigate parametrized complexity of the problem with respect to the structural parameters of the input graph such as minimum vertex cover, treewidth, tree-depth and neighborhood diversity as well as some intrinsic parameters of the problem such as the number of distinct star lengths, the maximum size of stars and the maximum degree of the input graph, giving a roughly complete picture of the parameterized complexity landscape of the problem.

cs.CC↗

arXiv2025

Halfway to induced saturation for even cycles

For graphs $G$ and $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$, and that $G$ is $H$-induced-saturated if $G$ is $H$-free but removing or adding any edge in $G$ creates an induced copy of $H$. A full characterization of graphs $H$ for which $H$-induced-saturated graphs exist remains elusive. Even the case where $H$ is a path -- now settled by the collective results of Martin and Smith, Bonamy et al., and Dvoŕǎk -- was already quite challenging. What if $H$ is a cycle? The complete answer for odd cycles was given by Behren et al., leaving the case of even cycles (except for the $4$-cycle) wide open. Our main result is the first step toward closing this gap: We prove that for every even cycle $H$, there is a graph $G$ with at least one edge such that $G$ is $H$-free but removing any edge from $G$ creates an induced copy of $H$ (in fact, we construct $H$-induced-saturated graphs for every even cycle $H$ on at most 10 vertices).

math.CO↗

arXiv2022

On the Parameterized Complexity of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to the parameters tw and (k, c4), where tw and c4 are the treewidth and the number of cycles with length 4 of the input graph. We also prove that the problem is fixed parameter tractable with respect to the parameter k for the line graphs and every proper minor-closed class of graphs (including planar graphs).

cs.CC↗