Picture of sea vessel plough through rough maritime conditions

Innovations in marine technology, pioneered through Open Access research...

Strathprints makes available scholarly Open Access content by researchers in the Department of Naval Architecture, Ocean & Marine Engineering based within the Faculty of Engineering.

Research here explores the potential of marine renewables, such as offshore wind, current and wave energy devices to promote the delivery of diverse energy sources. Expertise in offshore hydrodynamics in offshore structures also informs innovations within the oil and gas industries. But as a world-leading centre of marine technology, the Department is recognised as the leading authority in all areas related to maritime safety, such as resilience engineering, collision avoidance and risk-based ship design. Techniques to support sustainability vessel life cycle management is a key research focus.

Explore the Open Access research of the Department of Naval Architecture, Ocean & Marine Engineering. Or explore all of Strathclyde's Open Access research...

On graphs with representation number 3

Kitaev, Sergey (2013) On graphs with representation number 3. Journal of Automata, Languages and Combinatorics, 18 (2). pp. 97-112.

[img] PDF (Kitaev-JALC-2015-On-graphs-with-representation-number-3)
Kitaev_JALC_2015_On_graphs_with_representation_number_3.pdf
Accepted Author Manuscript

Download (315kB)

Abstract

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)$ is an edge in $E$. A graph is word-representable if and only if it is $k$-word-representable for some $k$, that is, if there exists a word containing $k$ copies of each letter that represents the graph. Also, being $k$-word-representable implies being $(k+1)$-word-representable. The minimum $k$ such that a word-representable graph is $k$-word-representable, is called graph's representation number. Graphs with representation number 1 are complete graphs, while graphs with representation number 2 are circle graphs. The only fact known before this paper on the class of graphs with representation number 3, denoted by $\mathcal{R}_3$, is that the Petersen graph and triangular prism belong to this class. In this paper, we show that any prism belongs to $\mathcal{R}_3$, and that two particular operations of extending graphs preserve the property of being in $\mathcal{R}_3$. Further, we show that $\mathcal{R}_3$ is not included in a class of $c$-colorable graphs for a constant $c$. To this end, we extend three known results related to operations on graphs. We also show that ladder graphs used in the study of prisms are $2$-word-representable, and thus each ladder graph is a circle graph. Finally, we discuss $k$-word-representing comparability graphs via consideration of crown graphs, where we state some problems for further research.