A characterization of uniquely vertex colorable graphs using minimal defining sets

Hajiabolhassan, Hossein; Mehrabadi, M. L.; Tusserkani, Ruzbeh; Zaker, Manouchehr

doi:10.1016/s0012-365x(98)00304-5

Cited by 7 publications

(5 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many papers concerning defining sets. We refer the reader to [6] and the survey paper [5]. The greedy defining sets in graphs were first defined in [10].…”

Section: Greedy Defining Sets In G Hmentioning

confidence: 99%

First-Fit coloring of Cartesian product graphs and its defining sets

Zaker

2016

Preprint

View full text Add to dashboard Cite

Let the vertices of a Cartesian product graph G H be ordered by an ordering σ. By the First-Fit coloring of (G H, σ) we mean the vertex coloring procedure which scans the vertices according to the ordering σ and for each vertex assigns the smallest available color. Let F F (G H, σ) be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether F F (G H, σ) = F F (G H, τ ), where σ and τ are arbitrary orders. We study and obtain some bounds for F F (G H, σ), where σ is any quasi-lexicographic ordering. The First-Fit coloring of (G H, σ) does not always yield an optimum coloring. A greedy defining set of (G H, σ) is a subset S of vertices in the graph together with a suitable pre-coloring of S such that by fixing the colors of S the First-Fit coloring of (G H, σ) yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of G H with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in G H, including some extremal results for Latin squares.

show abstract

“…There are many papers concerning defining sets. We refer the reader to [6] and the survey paper [5]. The greedy defining sets in graphs were first defined in [10].…”

Section: Greedy Defining Sets In G Hmentioning

confidence: 99%

First-Fit coloring of Cartesian product graphs and its defining sets

Zaker

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…A closer look at Force χ (2) is taken in Section 4. In Section 5, using a combinatorial result of Hajiabolhassan, Mehrabadi, Tusserkani, and Zaker [20], we analyze the complexity of a related graph invariant, namely, the largest cardinality of an inclusion-minimal forcing set. Before taking into consideration the forcing clique and domination numbers, we suggest a general setting for forcing combinatorial numbers in Section 6.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…Another related invariant of a graph G is the largest cardinality of an inclusionminimal forcing set in G. We will denote this number by F * χ (G). A complexity analysis of F * χ (G) is easier owing to the characterization of uniquely colorable graphs obtained in [20]. Proof.…”

Section: Maximum Size Of a Minimal Forcing Setmentioning

confidence: 99%

On the Computational Complexity of the Forcing Chromatic Number

2005

View full text Add to dashboard Cite

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number F χ (G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of F χ (G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if F χ (G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that F χ (G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if F χ (G) ≤ k, for each constant k, is reducible to a problem in US via a disjunctive truth-table reduction.Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

show abstract

“…in Fischermann [8], Fischermann et al [9], Fischermann and Volkmann [10,11], Gunther et al [13], Hajiabolhassan et al [14], Hopkins and Staton [18], Siemes et al [20] and Topp [22].…”

Section: Theorem 1 (Zverovich and Zverovich [25]) A Graph G Is Dominmentioning

confidence: 99%

Unique irredundance, domination and independent domination in graphs

Fischermann

Volkmann

Zverovich

2005

Discrete Mathematics

View full text Add to dashboard Cite

A subset D of the vertex set of a graph G is irredundant if every vertex v in D has a private neighbor with respect to D, i.e. either v has a neighbor in V (G)\D that has no other neighbor in D besides v or v itself has no neighbor in D. An irredundant set D is maximal irredundant if D ∪ {v} is not irredundant for any vertex v ∈ V (G)\D. A set D of vertices in a graph G is a minimal dominating set of G if D is irredundant and every vertex in V (G)\D has at least one neighbor in D. A subset I of the vertex set of a graph G is independent if no two vertices in I are adjacent. Further, a maximal irredundant set, a minimal dominating set and an independent dominating set of minimum cardinality are called a minimum irredundant set, a minimum dominating set and a minimum independent dominating set, respectively, and the cardinalities of these sets are called the irredundance number, the domination number and the independent domination number, respectively.In this paper we prove that any graph with equal irredundance and domination numbers has a unique minimum irredundant set if and only if it has a unique minimum dominating set. Using a result by Zverovich and Zverovich [An induced subgraph characterization of domination perfect graphs, J. Graph Theory 20(3) (1995) 375-395], we characterize the hereditary class of graphs G such that for every induced subgraph H of G, H has a unique -set if and only if H has a unique -set. Furthermore, for trees with equal domination and independent domination numbers we present a characterization

show abstract

A characterization of uniquely vertex colorable graphs using minimal defining sets

Cited by 7 publications

References 2 publications

First-Fit coloring of Cartesian product graphs and its defining sets

First-Fit coloring of Cartesian product graphs and its defining sets

On the Computational Complexity of the Forcing Chromatic Number

Unique irredundance, domination and independent domination in graphs

Contact Info

Product

Resources

About